I believe so, yes.

Prior to the mid-1960s, the vast majority of Americans did not have a state-issued ID that included a photograph. (Although US passports have included photographs since 1914, prior to the 1990s it is estimated that less than 10% of Americans held them; driver's licenses also only began being issued with photographs in the 1960s, with California leading the way.)

So if you wanted to open a bank account in the 1940s, you would provide things like utility bills, paystubs, corporate letterhead or noterized letters to establish your identity. When opening the account, you would sign a "signature card," which would be permanently retained by the bank to check your signature against if a question ever came up about your identity (e.g. someone forged a check on your account). You would also be issued with a passbook. When you went to the bank and completed a transaction, you would fill out a deposit or withdrawal slip, which the bank would keep for its records; the transaction would also be noted in the passbook, either by the teller's initials or a stamp of some kind, or both.

So if Andy Dufrense walked up to a teller's window, with the correct passbook for the account, and his signature matched the one on the signature card, he should have no trouble withdrawing all the money and closing the account.

What Andy would have to do as a prisoner to successfully open the account in the first place, I'm not 100% sure of. Presumably as an accountant he would be well aware or the requirements for opening an account and may have had access to notary stamps or letterhead as part of his work for the warden at the prison.