Doesn't have to be a name. They can quote department x and a code y which doesn't identify the employee. Once you call back, you can quote both and they can translate y -> employee name internally without letting the customer know any identifiable information about the employee.

I don't think this applies to calling back someone who just called you. Also people who have premium banking tend to use the same relationship manager every time without this suspicion.

That's a red flag on that bank.