One problem is that when the bank fails to adequately bet whoever they gave a loan to, it becomes your problem until you can convince them that they made a mistake (hope you can afford a lawyer). A standard ID card with a photograph would help if it meant that nobody would accept anything else (utility bills, etc. are much easier to steal or forge) and it would transform online banking if we had something like OpenID/U2F which would work online.

The other problem is that without a reliable primary key, it's much easier for correct information about different people to be incorrectly mixed. While this is not as bad as, say, when it hits a sex offender database or terrorist watch list but it does happen and would go down dramatically if everyone could just use something like a SSN as the link instead of trying various fuzzy heuristics and hoping for the best.