They don't need to know the requesting address in order for them to know if it was you the person corresponding to that proof of age, because the information is in the data being exchanged. These kind of verifications don't depend or rely on IP address or networking, these are credentials that are checked on the application layer.

In fact, they don't even need to directly communicate with the government for this.

This is equivalent to a registration office for a service asking you provide a paper stamped by the government that certifies your age without the paper actually saying who you are.. the service does not need to contact the government if they can trust the stamp in the paper and the government official signature (which in this case is mathematical proof). And even though the service office can't see your name in the paper, the government knows that the number written in the paper links to you individually, because they can keep record of which particular paper number was issued to which individual, even if your name wasn't written in the document itself.

So, the government can, at any given time, go to those offices, ask them to hand in the paper corresponding to a particular registration and check the number to see who it belongs to.

The traceability is in the document, not in the manner in which you send it. It does not matter if you send the document to a different country for someone else to send it from a different address, on your behalf (ie. a proxy). If the government can internally cross-reference the registration papers as being the ones linked to your governmental ID, they can know it's yours regardless of how it reached the offices. So this way they can check if you registered yourself in any particular site they wanna target and what your account is.