I was told that two loans, at different banks, can’t have the same account number?

I called today to do a balance transfer from a bank loan to a credit card.

The man only asked for my account number, and then he told me the name of the lender it was coming up under, which is not my lender. From Google searches, this non-bank company is not affiliated with my lender.

He then told me that two accounts (at different institutions) cannot have the same account number. Is that true? I can see how that could be true, (loan account numbers COULD be dictated by the Fed or something), but this isn't something I've ever known.

My concern is that if he's wrong, he's gonna pay some other loan with my money.