There's no such thing as identity theft. When Person A claims to be Person B and obtains a loan, he's defrauded the bank, not Person B. Person B is only harmed when the government fails to protect him from the bank that was defrauded. How so?
Person A has defrauded the bank, not Person B. The bank then harms Person B by going after him for the loan that Person A took out. The bank fell victim to Person A and then victimizes Person B. Person B is never a victim of Person A, so you can't say that Person A "stole" the identity of Person B. That language implies that Person B is a victim of Person A when he's not. Person B is a victim of the bank. The bank is a victim of Person A.
It is the government's job to protect Person B from the bank's claim that he took out a loan that Person A took out in the name of Person B. Person B should be considered innocent until proven guilty, and given due process, paid for by the bank when it loses it's claim. Instead, the bank is permitted to harass Person B and make his life a living hell, and when it is taken to court, Person B is left with all sorts of charges due to the bank's failure to verify that Person A was who he said it was. This is a failure of the bank and government in the protection of Person B's property.
If this were allowed to play out, I believe banks and other financial institutions would require stronger evidence for identification, etc. I believe this would be handled properly, with the bank assuming the burden of proof and costs of litigation should it fail to prove that Person B took at the loan, in a truly free market.