I suppose the word "system" is doing the bulk of the work here. An ID as a platonic thing is just something that uniquely identifies me, and we have an infinity

I suppose the word "system" is doing the bulk of the work here. An ID as a platonic thing is just something that uniquely identifies me, and we have an infinity of integers that would work just fine for that, SSNs being a subset of the integers too. As part of a system, though, this ID should not just only identify me and only me, but no one should be able to identify themselves with any other ID. You can probably classify this into a complete system as a form of authorization to identify, or a process of authentication to be done on ID usage, or a process of verifying an ID or ID derivative (like a signature) which may or may not require authentication. There are many ways of implementing this from photos to fingerprints to passwords to (my preference) smartcards that never expose the real ID and rely on asymmetrical encryption.