There is just more than a singular definition of identity in English and one of them can't be stolen while several others can. Impersonation is one way of stealing one of those definitions identity theft refers to. This doesn't mean the definition of identity is simultaneously violated.

The victim of impersonation isn't held accountable for the action of the criminal, particularly with banks. That's precisely what identity theft laws protect. I'm in favor of making that process even easier for the victim wherever possible but changing the name is not that.