If all the ID consists of, then no it's not.

As long as the part asking for ID trusts the part verifying the ID, there is no need for anonymity to be broken, since the verifier just has to confirm what the asking part needs to know.

Think of it like someone owns a bar and needs to know if a patron is old enough to drink, and the bar owners brother or best friend says "I know that guy, he is old enough".

Not necessarily. As another user noted, zero-knowledge proofs might be able to be used to anonymously age-verify people, if done correctly.