Zero knowledge proofs can perform expressions that check values within a JSON tree without exposing any of those values to the requesting party, for instance "year of birth < 2005" can return true or false without returning the person's numeric birth year. Essentially the requesting party has the holder of the credential perform a computation, the result is guaranteed to be the result of each and every instruction over a target data structure (only knowing the hash and signature chain of the credential, so for instance your government issued id can be signed by your secretary of states public key)
Estonia has a really interesting government issued public key infrastructure where users can validate their identity with their physical ID card and a USB reader (maybe it's NFC by now?) but I don't think I've heard of the above scheme used in practice, just sat through a presentation at the internet identity workshop.
Zero knowledge proofs based on too little information are trivial to abuse.
To combat this, you need to have it based off of more and more personal info....which is at odds with the privacy-preservation goal.
Sadly when it comes to age assurance, Zero knowledge proofs are little better than marketing.
But the verifying party can still track you because they can (and absolutely will) log who the requester was and when it was requested. The site might not know who you are, but the government will now have a record of all your 'adult web activity'.