In the ZKP system Europe will be using and I believe in the one Google has developed when you verify your age to a site the communication is only between your device and the site.

The only information the site gets that they don't get when you login now without any kind of age verification (other than something like clicking "I am 18+") is that you have a government issued ID that says you are 18+.

If their logs without age verification are insufficient to reveal who you are if they get turned over to the government then the logs with age verification will also be insufficient.