Maybe we don't need a unique id at the federal level after all. Financial institutions are already required to follow KYC (Know Your Customer) laws. For this they need a way to validate the identity of the customers they do business with. I guess they already have enough ways to do this without relying on SSN or a potential national id card.
BTW, after the Equifax hack, financial institutions should be mandated to stop trusting SSN as proof of identity.