Suppose a natural-born citizen of one of those countries principally objects to the notion of monarchy and in particular is unwilling to swear allegiance to a king or queen - are there any rights or privileges that would be denied to them such as getting an ID, a passport, a drivers license etc. ?
What about a non-natural person otherwise qualifying for citizenship or a passport?
TLDR
From natural-born citizens at least, no, you don't need to swear such allegiance, unless you're doing some very specific jobs. Even in such jobs, possible with some exceptional circumstances, one can be 'anti-Monarchist' without repercussion.
I'm British, so I can comment on the "natural-born" part of this question. What I say below applies for such people. I don't know about people applying for citizenship "from outside", but I expect similar claims hold.
While personally I have no issue with the royals, I know lots of people who oppose them, some very strongly and openly; I don't remember specific names, but if you look up famous British comedians, say, you'll see that a lot of them go on these panel shows and, if it comes up, have a rant about how much they hate the concept of 'royals'.
@LightnessRacesinOrbit points out in the comments, that Jeremy Corbyn, the leader of the opposition party, is such a person. Although he claims that he wouldn't try to remove the monarchy were he in power (Independent article). I guess he feels there are more important things; but this doesn't restrict him from becoming PM (prime minister), and even as the PM he would meet the Queen weekly -- even if he were to campaign for abolition of the monarchy, he'd still meet her weekly! (Might be awkward...)
Personally, I don't recall (in my 24 and a bit years) ever being officially asked about this. In particular, certainly for a passport or driver's licence I never needed to. As mentioned by origimbo, to be an MP (member of parliament) you have to swear an oath to the monarch, but that's a pretty specific thing -- much more so than getting a passport. Similarly, police officers swear an oath to serve the Queen (described and cited here) -- although @inappropriateCode points out below that this is not the case in Northern Ireland (which makes sense, culturally).
I think in general the idea that you are allowed to have your own political view is held well above the idea that you have to like the monarch. As the saying goes, "it's a free country".
I have been a Canadian my whole life and I do not recall ever being asked to "swear allegiance" to the monarch verbally.
Generally the monarchy is popular in a celebrity kind of way and while there are some anti-monarchists I believe it is not a significant political issue.
Since individual rights are important to Canadians (and I imagine similarly to the people of the U.K. and Australia) political views are not a hindrance to living a normal life for the vast majority of Canadians.
At this link are people who are openly anti-monarchy and I would guess they have drivers' licenses and passports.
New Canadians have to declare fealty to the monarch when they make Canada's Oath of Citizenship.
This is a complete non-issue in the UK. For example, the leader of the opposition in Parliament, Jeremy Corbyn, has openly refused to sing the National Anthem when attending high profile, televised, public events such as a memorial service commemorating the anniversary of the Battle of Britain in WWII.
It's hard to imagine any public figure "getting away with" that sort of behaviour in the USA!
Of course many people (including Corbyn, going through the formalities of parliamentary procedure) may sometimes do things that they don't believe in - but the vast majority of the UK population is never forced to make any declaration of allegiance to, or show any respect towards, the monarchy, the national flag, and similar national symbols.
British person here, swearing an oath of loyalty to the monarch only happens if you join government service of some kind (millitary, police, etc). Otherwise it almost never comes up. While I quite like the royals, a lot of people don't like them and are free to do so.
Others have discussed most about those having citizenship from birth. I will attempt to answer this part of your question:
What about a non-natural person otherwise qualifying for citizenship or a passport ?
To become a UK citizen, one must take an oath of allegiance. The text of the oath (from Wikipedia) is as follows:
I, [name], [swear by Almighty God] [do solemnly, sincerely and truly affirm and declare] that, on becoming a British citizen, I will be faithful and bear true allegiance to Her Majesty Queen Elizabeth II, her heirs, and successors, according to law.
This oath is to be taken in one of the final steps in becoming a citizen, during a so-called citizenship ceremony.
Other Commonwealth countries have similar requirements. For example, that first Wikipedia page also lists the oath for future Canadian citizens:
I swear (or affirm) that I will be faithful and bear true allegiance to Her Majesty Queen Elizabeth the Second, Queen of Canada, Her Heirs and Successors, and that I will faithfully observe the laws of Canada and fulfil my duties as a Canadian citizen.
Or in French, of course:
Je jure fidélité et sincère allégeance à Sa Majesté la Reine Elizabeth Deux, Reine du Canada, à ses héritiers et successeurs et je jure d'observer fidèlement les lois du Canada et de remplir loyalement mes obligations de citoyen canadien.
The other answers comprehensively cover the UK and Canada cases. I'll offer the Australian perspective, as a naturalised Australian citizen.
Contrary to Canada, the Australian citizenship pledge does not pledge allegiance to the Queen. It it a common misconception though, so much so, that when one of my friends is about to become a citizen, I tease them by asking: "are you ready to offer your allegiance to the Queen?". They are pleasantly surprised when they learn the actual text. There are two versions of the pledge: including the phrase "under God", or not.
From this time forward, [under God,] I pledge my loyalty to Australia and its people, whose democratic beliefs I share, whose rights and liberties I respect, and whose laws I will uphold and obey.
The current version of the pledge was introduced in 1994. Older versions did offer allegiance to the King or Queen.
Moreover, the members of the Parliament and members of the Armed Forces still have to make an oath (or affirmation) that includes allegiance to the Queen. But in practical terms, I do not believe that this stops them from expressing anti-monarch views and sentiments.
A fact that supports this belief is that the previous prime minister of Australia, Malcom Turnbull, came to political prominence in the late 90s leading the Republican movement in Australia. In other words, an anti-monarchist politician reached the highest political position in the country.
H2ONaCl's answer gives a good Canadian perspective, there is a similar list for the UK here, which lists Members of Parliament, prominent journalists and media personalities.
MORI polls cited on the Wikipedia page show around 20% of those sampled expressing republican views.
The current Leader of the Opposition (Jeremy Corbyn) is one of the people mentioned, so it's not only possible, but quite a popular viewpoint in the UK.
It's true that Members of Parliament swear an oath to "Her Majesty Queen Elizabeth II, her heirs and successors, according to law", but this has been interpreted as an oath to the person and not the office, or with the emphasis on "according to law", which does not preclude holding republican views. There are some more (or less) artistic variations on this idea from republican politicians listed here.
The other answers clearly address the practical aspect of the question, namely they correctly suggest that it is possible (and common) to live in the Commonwealth while openly disavowing the monarchy. Nonetheless, the formal legal situation may be much less favorable.
Indeed, under the Treason Felony Act 1848 of the United Kingdom, it is treason felony to "compass, imagine, invent, devise, or intend" to deprive the Queen of her crown in any of her dominions and countries (whether or not you are located in the United Kingdom yourself). The maximum punishment for this crime is life imprisonment.
This section of the Wikipedia page provides some discussion of the Guardian's attempt and failure to challenge this Act in 2001, as well as some more activity in 2013 that indicates that the law is still in force. See "Calling for abolition of monarchy is still illegal, UK justice ministry admits" in the Guardian.
The last charges under this act appear to have been in 1972. Those charges were reduced to the lesser charges of seditious utterances, which itself was a crime in the same vein. This charge appears to have been abolished, except for aliens, addressing another aspect of your question.
It is worth noting there are certain circumstances, that a non-natural citizen can acquire British citizenship without swearing the Oath of Allegiance. As I was a minor when I acquired my British Citizenship, my mother didn't want me to skip school (as she could get fined for doing so) to attend the ceremony and informed the council that she would be picking up my certificate as well as her own, which they were perfectly happy to do.
There seem to be at least three different issues being discussed here.
One is, essentially, whether you are obliged to accept the authority of the law, the constitution, and the government of the country, which has the monarch as its titular head. The answer is: yes, you are. If you behave in a way that flouts that authority then you will end up in prison.
Two is whether you are obliged to say that you like this system. Answer: no, you aren't. Even those who swear allegiance are only agreeing that they will go along with the system, not that they actually like it or are prepared to defend it.
Three is whether you like the individual currently holding office. That's almost totally irrelevant. Except that the British system is actually quite flexible, and there have been quite a few occasions over the centuries that a monarch, or proposed monarch, has been unacceptable to the people (or the barons) and has been quietly sidelined. It could happen again.
In the 1990s, if an Australian boy or girl became a Scout, they had to swear "to do my duty to my God, and to the Queen of Australia." However these days they can choose instead to swear "to be true to my spiritual beliefs, to contribute to my community and our world."
