This thing has a masking step, so you always prove to any particular site that you're the same person you were the last time you visited. But you don't (deliberately) present them with any clue as to who that is exactly.

There's even a deliberate choice where though you can specifically identify models of product [ e.g. "This is a Mattel Barbie brand authenticator" so that in principle a bank could agree to use this standard but they only want to accept official bank-branded tokens ] the specification says DO NOT put unique serial numbers inside the tokens, if you want to distinguish them, make sure only large runs are distinguished e.g. "Batman authenticator" versus "Barbie authenticator" but never distinguish "Steve's authenticator" versus "Dave's authenticator" so as to preserve the privacy mechanism.