My bank actually gets this right. They very, very rarely call me, but when they do it's
"Hello, am I talking to Nick Lamb?"
"Yes, this is me"
"OK, I'm calling from Example Bank and our confirmatory password is Melons" [not the actual bank or password]
"Thanks, that checks out, what can I do for you?"
This happened because I had one of those conversations you're talking about, and they were like "Aha! We have something we can do for those situations, call us and set a password we can use" so I hung up and sure enough they've used that password ever since. I like it.
It's not a _good_ password, but hey, how many times does anyone try the wrong one? Literally never. So it's good enough.