The problem isn't the banks, the problem is that unlike Europe where in most countries it's commonplace for everyone to have a government-issued ID document, the US does not have that requirement and so companies of all sorts abuse documents not meant for that purpose like SSNs or driver licenses that can be trivially forged.
Banks can't invent security out of thin air when a significant part of the US population believes that mandating possession of one is a surefire way into a dictatorship or whatnot.