Zero knowledge proof is either trivially defeated by re-using the same credentials or doesn't have useful privacy guarantees. There really isn't an in-between here for something like age verification.
And you don’t see a problem with this part?
What the ZKP does is let you limit the information the site collects to the fact that you are under 18, and nothing else. It’s an application of the principle of least privilege. It lets you give the website that one fact without revealing your name, birthdate, address, browsing history, and all your other private data.
After all - if it doesn't share anything other than a guarantee of the "age" of someone who is authenticating with the website then how would the website know there's re-use of identifiers?