Speaking as a citizen of a country with "real" identification laws, how would you like the banks to identify you?
SSN? Absolutely not. That number is not meant to be used as formal identification, it's completely unprotected.
America has a particular weakness to identify fraud because you don't have an identity system.
My ID number is formally defined in law, protected and verified by various check sums built into the number (like so many tokens out there). Banks are required, by law, to ensure they have identified the right individual before transacting and they carry the full burden if they get it wrong.
The SSN in the US is sequential and I never fail to laugh at the idea of forcing your banks to verify you with such a flimsy structure. It's just gonna make the problem worse