TL;DR: multiple questions here:
How do international news agencies make their maps of Russia's military progress in Ukraine
There are press briefings, and some of these are accompanied with data and electronic media. Government agencies have hand-outs and provide information to assist in getting a narrative out
Independent think tanks and institutes and agencies are dedicated to these topics and they provide products in the form of maps that news agencies use and also reinterpret using their own resources.
and where do they get their data?
Using the UK Ministry of Defence as an example they have many agencies tasked with collecting and analysing data. Many have historic links back to when SIS was making covert reconnaissance missions into and over occupied territory in previous wars. Together they collect human, signal, geographic, aeronautical, geospatial, satellite information, analyse and then share the results of this information throughout the relevant agencies and allied partners. One part of that is used to produce an info graphic product that today you can see on twitter, for example.
Think tanks are often closely related to the military and whilst not always having access to the same information can certainly make more educated guesses to build products such as info graphics and maps.
In both cases OSINT (open source intelligence) is extremely important today and even those outside of these systems, just amateurs at home are using the same available data to build their own maps.
But back to the point, the ones you see on news agencies such as in the OPs question: Thank tanks such as ISW and self-published by MoD and if not directly reprinted then they are re-interpreted for the agencies own editorial usage.
Further example is the French Ministere des Armees self-publishing its own maps via its own website and twitter. (its probably worth noting the variations between different countries versions of the maps)
https://www.defense.gouv.fr/ukraine-point-situation
"bonus points" for what the red areas actually denote.
Red says Russian controlled, see map and link below
I wonder if they are all simply copying each other?
A few lesser outlets might do, most licence. The major agencies would be producing their own though, using the sources outlined above.
Or are they getting it from some commercial source, perhaps a satellite imaging company that offers analytical services as well?
All of the above, companies such as Maxar and Planet /Lab are mentioned below. Both Maxar and Planet operate over 200 satellites each. What is new is the increasing use of amateur/crowdsourcing/citizen-journalism/social media for data/mapping purposes
Or are there state-controlled outlets that offer this kind of information to the press as background?
Yes. As shown below. (Again, the twitter page for the UK MoD is a great example)
.........
Too long segment:
Combination of:
Hand outs from government agencies, usually, as primary source.
(Note: the maps distributed are often 'infographics' themselves, vetted, and then get reinterpreted as infographics in the news outlets)
The news agencies (SKY for example) would then have their own internal departments redraw them out in friendly format (infographics). Sometimes this is farmed outside as well. Generally, as these are civilians and they could be doing the same work for road traffic closures as much as a battlefield their background in (and technical accuracy therefore) is sometimes limited. I have seen rather inaccurate maps flying about.
Secondary is the news agencies own staff would be collecting data on the ground or from other agencies and compiling it (same when not in war scenarios).
And then there are civilian agencies, private, and I include think tanks, usually composed of ex-military, who are doing these all the time and then these get handed out as requested.
As they are ex-mil or are specialised in this area their accuracy is usually greater and they have the background knowledge to complete the tasks to a greater degree, yet also know how to make it civilian friendly.
Example:
Look at the maps and the sources for each map presented:
https://www.nytimes.com/interactive/2022/world/europe/ukraine-maps.html
Source: Institute for the Study of War with American Enterprise Institute’s Critical Threats Project
https://graphics.reuters.com/UKRAINE-CRISIS/zdpxokdxzvx/
Source: Janes; Institute for the Study of War with AEI’s Critical Threats Project
https://edition.cnn.com/2022/02/24/europe/ukraine-visual-explainer-maps/index.html
Source: Institute for the Study of War, Google Maps, Maps4News
The data is re-interpreted and re-presented as an infographic, but the source in each case is from the typical agencies listed later in this answer.

You also have press briefings that are accompanied by handouts, digital and analogue:


(Here the journalist has generated an infographic based on the press briefing information)
....
The government departments obviously have their own sources from within the military that dictates what they can and cannot show.
External agencies, think tanks, often provide the most seen source of map/infographic data in the news:
The external agencies use a combination of both government and their own sources (some open/public), but abide by what the military say cannot be shown (shown is Russian forces only for example).
Example of typical external agencies generating maps and data:
https://rochan-consulting.com/
https://www.understandingwar.org/
https://www.criticalthreats.org/
https://www.janes.com/
https://www.csis.org/
https://www.iiss.org/
These agencies often work with other companies for satellite data (given that the government would have access to its own intelligence gathering hardware/software):
For external private map analysis, Maxar Technologies is a name I have seen frequently (Planet Lab also):

Agencies like these use their resources (hardware like satellites, software and human analysts) to offer these end user products to government agencies, private sector and media outlets:
https://www.maxar.com/products/crows-nest
Maxar’s Crow’s Nest Maritime Monitoring and Security capabilities leverage space-based optical and radar imagery and advanced machine learning to deliver critical information quickly
https://www.maxar.com/who-we-serve/international-civil-government
Maxar supports local agencies and the humanitarian community by providing critical information to assist response efforts.
...
The following think tank is the most currently visible source that I have seen for Western media map/infographic data when regarding the current Ukraine situation:
THE INSTITUTE FOR THE STUDY OF WAR
ISW’s recent maps of Russian military operations in Ukraine have reached millions of people and have been featured in prestigious print and broadcasting media and are used by humanitarian organizations.
ISW synthesizes multiple open-source information with data-driven analytic software platforms such as ESRI ArcGIS that support investigations enabling production of educational graphical maps used by many agencies.
(Is the source for both maps seen in the CNN and Sky news links of the OP's question, the ISW is a United States–based think tank)
https://www.understandingwar.org/backgrounder/russian-offensive-campaign-assessment-march-16

The Critical Threats Project, from the American Enterprise Institute is dedicated to tracking and analyzing key and emerging national security threats to the United States in order to inform the policy debate, is typical of many enterprises that collect and provide information for both private/commercial and government sector agencies.
Working with many different agencies (including ISW, above) to 'build' their product this is then sold (or licensed, or made freely available) and is used by the end user.
https://www.criticalthreats.org/locations/ukraine
Example of continuous map updates available:

Or are there state-controlled outlets that offer this kind of information to the press as background?
Example:
Straight out copy from Ministry of Defence in UK to a UK national newspaper (I didn't match the dates):

The twitter page for the UK's Ministry of Defence is frequently releasing map infographics which are then used by news agencies to illustrate their stories.
(though with the DoD less so)
....
As for the SKY news question:
this is the relevant infographic with legend:

https://news.sky.com/story/russias-invasion-of-ukraine-mapped-what-happened-on-day-eight-12556318
(The SKY map is based on ISW infographics. ISW classifies (red) areas as 'assessed Russian-controlled' - the assessed bit is important: when information is assessed and it is subsequently deemed to be of value then it becomes intelligence. So a judgement call has been made by analysts that says this area for example has Russian troop presence making Ukraine troop presence infeasible. Logistically Russians are keeping to the roads, which means if the roads (especially in close proximity) have their presence the area gets marked red, but would also not necessarily mean that Russians even occupy the region of land in between those roads either. It does not mean Russians on every corner or that there are zero Ukrainians present. Similarly Assessed advances are just where Russians have been seen in but are not in an occupying position, ie. they may have launched an attack but are not necessarily still in the area. Remember these are greatly simplified for easier consumption, and any real map of the situation would be incredibly complicated, for obvious reasons.)
...
Maps and/or infographics in todays news, in general:
News agencies internally would use software such as this for creating and annotating maps based on the information they have received from government agencies:
https://maps4news.com/gallery/
create and publish beautiful basic story locators or to make map visualisations of your data

From an article linked at the end:
Where press badges and notepads were once the common tools of the trade, today’s journalists are just as likely to be found behind a computer, scrolling through rows of data in spreadsheets and databases.
The proliferation of data and the rise of digital media has enabled new techniques and channels to engage readers.
Recognizing that data fluency is now critical to journalism, outlets like the New York Times have even taken to hosting data bootcamps in their newsrooms to help reporters and editors on every desk become more comfortable navigating, analyzing, and understanding data.
But data fluency does not matter if readers can’t understand the analysis results or relevance. Enter the maps.
Free mapping tools for journalists:
Use maps as a storytelling tool to explain context around a breaking news event and create an immersive experience for your audience
https://www.journalism.co.uk/news/9-free-mapping-tools-and-resources-for-visual-storytelling/s2/a563094/
Article on the subject at hand:
Power and Responsibility: Maps and Journalism
https://www.directionsmag.com/article/8799
Hurricane Andrew in 1992 ...NOAA and NASA provided data and imagery to local and regional news outlets in near real time.
used GIS (Geographic Information System, mapping software) to illustrate how decisions made at the administrative level led to increased and preventable damage from this event. This GIS-based reporting earned him the Pulitzer Prize in 1993. This was the first recorded deployment of GIS in the newsroom, but certainly not the first use of mapping in journalism.
The Defence and Security Media Advisory Committee is an example of the buffer between government agency and the media, mapping data being a graphical example of that kind of material:
https://www.dsma.uk/
The aim of the DSMA-notice system is to prevent inadvertent public disclosure of information that would compromise UK military and intelligence operations and methods; or put at risk the safety of those involved in such operations; or lead to attacks that would damage the critical national infrastructure and/or endanger lives.
If you look at the members section, you can see it is made up of the major media outlets, and is an advisory body which oversees a voluntary code which operates between the government departments which have responsibility for national security and the media.
Example of up to date cartographic data generation:
At the UK Defence Geographic Centre:
https://www.gov.uk/government/news/keeping-the-uk-military-on-the-map
Producing infographics is a job, this is from a major news agency - I include it to demonstrate this would be an internet department producing the maps and graphs and motion graphics that you are presented with every day:
Summary:
Creation of designs and graphic assets representative of *** brand values. Playing a crucial role in the visual design formulation and creation process and staying ahead of the curve with current trends. Solving information design challenges and adding clarity to ***'s array of output. (Work well in collaborative, dynamic teams of producers and editors
Ability to conceptualize and bring creative ideas to projects
Create 2D and 3D custom animated graphics for social media platforms; etc)
These days, everyone is looking at eachother - I suppose an example of just copying:

....
Further reading:
https://richybedford.wordpress.com/2014/01/07/an-academic-study-into-the-use-of-gis-in-journalism/
Crowdsourcing in journalism to build maps:
https://youtube.com/ watch?v=HRNnz4LJvx0&feature=share
https://richybedford.wordpress.com/category/maps-gis-and-spatial-data/
Where can journalists get GIS data?
https://richybedford.wordpress.com/2014/01/08/where-can-journalists-get-gis-data/
A mapping toolbox for journalists
http://mediahack.co.za/2017/05/mapping-toolbox-journalists-10-tools-worth-checking/
Guardian guide to making an infographic:
https://www.theguardian.com/global-development-professionals-network/2014/aug/28/interactive-infographics-development-data
The essential lies in news maps
https://datajournalism.com/read/longreads/the-essential-lies-in-news-maps
Newsrooms have also discovered the power of before-after satellite images. Often, the supplier of these images is Planet, a Silicon Valley satellite imaging company that offers daily high resolution pictures with its 'flock' of shoebox sized satellites.
How Maps Help the Media See What Others Can’t
https://www.esri.com/about/newsroom/blog/how-maps-help-the-media-see-what-others-cant/
The Rise of OSINT:
https://www.theweek.co.uk/news/technology/956029/what-is-open-source-intelligence-ukraine-war
Open-source intelligence (OSINT) researchers have “have existed on the fringes of conflicts since at least 2014”. Operating “across the world”, these researchers “publicly conduct the type of work that intelligence agencies do behind closed doors”.
Since the Cold War, the US and its Nato allies have tracked Russian troop and equipment movements “using expensive and often exotic means of keeping tabs on other people’s territory such as spy satellites and surveillance flights”, The Economist said. But now, “journalists, academics, think-tankers, activists and amateur enthusiasts” are increasingly ... using “their own tools” to “reveal goings on in inaccessible places”.
https://www.freethink.com/technology/osint-ukraine
What is unfurling across Twitter and other social media spaces is the power of publicly available information. Known as open source intelligence, or OSINT, it can be viewed by anyone, if they know where to look, and analyzed by anyone, if they know what to look for.
a combination of footage captured by private companies and individuals means that a wealth of information about a conflict is available to anyone willing to look for it — and that includes governments.
“Government OSINT typically provides 90% of the intelligence used for policymaking and national security related decisions,”..
“This situation is no different, especially since the U.S. government is using collection from a variety of non-government open sources to gain situational awareness, understanding, and confirmation of Russian intent and military posture, including damage inflicted on Ukrainian interests.”
https://www.nbcnews.com/news/world/putin-russia-ukraine-invasion-tiktok-biden-nato-rcna13795
Professional analysts and amateur sleuths have turned to a mix of social media videos and satellite imagery in an attempt to gain insight into the Kremlin’s plans.
https://www.oii.ox.ac.uk/news-events/news/did-libyan-crisis-mapping-create-usable-military-intelligence/
Did Libyan crisis mapping create usable military intelligence?
OSINT map showing movements of both sides in Ukraine:
https://twitter.com/JominiW/status/1503581349067866119?s=20&t=pD-V47sv6XPk6Ujkgxi_jA
Russian OSINT:
https://dragon-first-1.livejournal.com/52143.html?utm_source=3userpost