Fraud. It's called fraud. Someone is defrauding the bank. The bank is the victim. However, the person whose identity was referenced by the criminal has nothing to do with the interaction. The criminal did not steal an identity. They stole money from a bank through fraud.
That term was already coined as a synonym. I wonder which form of fraud does not involve some kind of misrepresentation though. What makes it notable that a person who actually exists is being used as the conman's fictitious identity if we aren't interested in somehow offloading the risk to that person? Presumably this happened before the 1960s when it began to be called "identity theft" (and if you look at an ngram the term really only comes to be widely used in the mid-to-late 90s).
