Most age verification services use either government providers or 3rd party providers. I show my passport (or whatever) to the third-party. They relay to the site "this user is / isn't over 18". They don't send the DoB, address, photo etc.
So the online service only receives a binary yes/no and nothing else. I don't lose any privacy there.
The third-party knows that you wanted to be verified on service xyz, but not what you do there. Depending on the service I'm using, I may or may not care that they know.
Handing over a passport / licence to get into a bar leaks more information than that.