Sadly, financial institutions will continue to use knowledge of your SSN and DOB as proof that you are who you claim you are. And if you're not, that's the problem of the sucker whose identity got stolen.
Financial institutions in America prioritize convenience over security.
What's better is congress wants to tackle piracy (which will never be stopped) by frivolous bills like SOPA, and make backdoors for encryption to "catch the terrorists / bad people" but nobody wants to fix identity theft. Heck, now we're all having to have stupid cookie dialogs on every website.
You don't need to change the core identifiers. You just need to stop treating (at an institutional and broader system level) mere knowledge of those identifiers alone as sufficient proof of a user's authenticity. For the most part, the 50-year-old hard-to-change code is already surrounded by other systems which can be adapted more easily anyway.
This is the real and hard problem to solve. As far as I know, there are identity-verification services using other, semi-publicly-available data, which can still be spoofed for a lot of people, and some that use just-in-time photography (of your face, driver's license, passport, etc.), but that relies on more on-device security (and thus less end-user ownership of their devices).
It ultimately falls to the government to provide a more robust solution.
Well, then these institutions can take more responsibility when their weak auth is exploited to defraud innocent people, vs "sucks to have your 'identity' stolen!"
Financial institutions in America prioritize convenience over security.