What's the problem with that? We have cameras, passwords, tokens, witnesses. Tell credit issuers to demand photo of person holding up ID and a thumbs up instead of just typing in a social security number. If that still results in too much fraud, then the lender is incentivized to do more research and verify in person that the person is who they say they are.
Right now, in the interest of increasing commerce, we're forcing everyone in society to bear the burden of this specific fraud and the benefits go to the lenders. If person A is being defrauded by person B pretending to be person C, then person C has no reason to be involved unless they aided person B.