The site would only know that the user's age is being vouched for by some government-approved service. It would not be able to use this to track the user across different devices/IPs, and so on.

The service would only know that the user is requesting that their age be vouched for. It would not know for what. Of course, they would have to know your age somehow. EG they could be selling access in shops, like alcohol is sold in shops. The shop checks the ID. The service then only knows that you have login credentials bought in some shop. Presumably these credentials would not remain valid for long.

They could use any other scheme, as well. Maybe you do have to upload an ID, but they have to delete it immediately afterward. And because the service has to be in the EU, government-certified with regular inspections, that's safe enough.

In any case, the user would have to have access to some sort of account on the service. Activity related to that account would be tracked.

If that is not good enough, then your worries are not about data protection. My worries are not. I reject this for different reasons.