I assume you're asking about value added tax (VAT) mostly, that tax is applied to goods and services at the final point of sale. With free movement of goods, I don't there are regular taxes for moving goods from one member state to another, whether that's within one company or as a business to business transaction.
In the EU, VAT is charged in the country of consumption. For example, when a Dutch customer orders something from amazon.de in Germany then they will pay the Dutch VAT rate. According to the EU's page on VAT:
For EU-based companies, VAT is chargeable on most sales and purchases of goods within the EU. In such cases, VAT is charged and due in the EU country where the goods are consumed by the final consumer. Likewise, VAT is charged on services at the time they are carried out in each EU country.
Regarding cross-border VAT, there's another page on the EU website that says:
If you sell goods and send them to consumers in another EU country, you usually need to register your business there and charge VAT at the rate applicable in that country - unless the total value of your sales to that country within the respective tax year falls below the limit set by the country.
With different tax rates and many EU member states that may seem a lot of work, especially if you don't have that many EU customers from certain countries. That's what the last sentence in the previous quote refers to, when sales from your business to consumers in another EU country remain below a certain threshold then you don't need to charge the VAT of that country. When that happens the business charges VAT in the country where it's registered.
In recent years, there's been a move to modernize VAT for cross-border e-commerce. The primary concern lies with creating a level playing field. You don't want a situation where businesses located in one member state (or even outside the union) get most of the benefits to the detriment of businesses in (other) member states.
I am not an expert at this, so I will simply quote the changes mentioned in the EU's announcement on the New VAT for e-commerce rules which apply since July 1st 2021 which is being introduced with upbeat claims like: "Cross-border VAT e-commerce is being modernised in the EU." and "We are making life simpler and fairer for all.". The following is quoted from the announcement after collapsing the 'what is changing?' tab:
From 1 July 2021, the VAT rules on cross-border business-to-consumer (B2C) e-commerce activities will change. The rationale for these changes is to overcome the barriers to cross-border online sales and address challenges arising from the VAT regimes for distance sales of goods and for the importation of low value consignments.
The main changes are the following:
Online sellers, including online marketplaces/platforms can register in one EU Member State and this will be valid for the declaration and payment of VAT on all distance sales of goods and cross-border supplies of services to customers within the EU.
They will benefit from a reduction in red tape of up to 95% by registering with the new One Stop Shop (OSS).
The existing thresholds for distance sales of goods within the EU will be abolished and replaced by a new EU-wide threshold of EUR 10 000. Below this EUR 10 000 threshold, the supplies of TBE (telecommunications, broadcasting and electronic) services and distance sales of goods within the EU may remain subject to VAT in the Member State where the taxable person is established.
Special provisions are introduced whereby online marketplaces/platforms facilitating supplies of goods are deemed for VAT purposes to have received and supplied the goods themselves (“deemed supplier”).
In addition, new record keeping requirements are introduced for online marketplaces/platforms facilitating supplies of goods and services, including where such online marketplaces/platform are not a deemed supplier.
The VAT exemption at importation of small consignments of a value up to EUR 22 will be removed. This means all goods imported in the EU will now be subject to VAT.
But help is at hand! A new special scheme for distance sales of low goods imported from third territories or third countries will be created. The Import One Stop Shop (IOSS) has been created to simplify the declaration and payment of VAT.
Finally, simplification measures for distance sales of imported goods in consignments not exceeding EUR 150 will be introduced, in case the IOSS is not used (special arrangements).
The above addresses the following issues (not an exhaustive list):
Businesses don't have to deal with tax authorities in different member states. As the announcement says, businesses can register in one country and declare and pay VAT from there.
All member states will have the same threshold for distance sales (that's the threshold below which they don't have to charge VAT following the rules at the point of consumption).
The 22 euro VAT exemption for importing goods (by consumers) into the EU has been abolished. This is aimed at the import of cheap goods by EU consumers in a way that allowed them to avoid paying VAT in the EU while not paying in the sending country either. That exemption put EU companies at a disadvantage because they had to charge VAT leading to higher prices for their goods.