Serial number scanning is a common feature of electronic cash counters (also networking!) I'm pretty sure all incoming serial numbers are scanned and checked against a database to find stolen banknotes. I don't know if any banks are tracking outgoing serial numbers, but it would be pretty trivial to implement. I've noticed when I go in and withdraw cash from a human teller that the cash no longer comes from a drawer, but rather is dispensed by an automatic cash handling machine and then handed to me.

Is the probability that large for bank1 vs bank2? There’s only a few major banks left. Chase has so many ATMs it’s hard to compete with the access.