The source behind that comment doesn't verify the claim that your browsing history is being shared. Only that the app currently being developed is a temporary app for use until the full app has been finished.

In fact, the linked article links to the EU website where it is specifically stated that the final protocol will be compatible with the temporary app (the anonymous age verification protocol).

The final app will also serve as a method to identify oneself (i.e. to a police officer) but that's separate functionality from the token based authentication.

The current lack of zero knowledge proofs does pose a potential privacy issue when websites and governments work together to track you across a length of time and re-authentications, but it's not like you're inherently sharing your browser history with the government. As far as I know, the temporary app intents to implement ZKPs but ran into standardization issues, so it's not like this is an intentional shortcoming either.