The party that attests to your age would still not know where you were using that attestation using regular digital signatures.
And yes, as previously mentioned the party you are testing to does not know your birthdate or any other information.
There is zero need for zkp with either of these
If the signer keeps a copy of the signature and who they made it for, someone who gets a hold of that and the records of party you used the signature at they can find out who you are.
There are ZKP based protocols that allow for age verification where even if the party that attests to you age keeps records they do not find out where you are using that attestation, and the party you use the attesting with only finds out that you are above their age threshold and what attesting party you used.
I think that this can be done without ZKP if instead of simple signatures we use blind signatures.