A weird one that happened to me a few years ago was when I ordered about $100 worth of products from banggood, my phone rang within seconds of the purchase going through. The display on my phone showed it to be a bank I never heard of. When I did a search, it turned out to be a small bank with about a dozen branches in a state hundreds of miles away. The voicemail however was from my bank's fraud department informing me they were locking my card due to unusual activity. When I called my bank using the number on my card, they verified that the original call was from them and they were able to unlock my card. They had no idea why the name of a different bank showed up on my phone. Best I can determine is that my cell provider was the one who added the name information and something in their database was incorrect.
I've definitely noticed that some small regional banks make extra money by offering services to other small banks and credit unions. For example, cashier's checks issued by my (fairly large actually) credit union are actually drawn on a midwestern regional bank that they seem to use as a service provider. So it's possible that bank is somehow involved in your bank's card issuance.
I also wonder if it's a contract call center or fraud prevention company and the caller ID value was just bad... at some point their phone number may have gotten put on a listing for one of the banks that uses them and now it's stuck in the Google Caller ID database or whatever. Google's caller ID is frequently pretty out of date and it's unclear how you would get it fixed (e.g. my husband has had his phone number for years and Android phones still show someone else's name when he calls).
Most banks don't do their own everything. A few seemingly little banks (often in the middle of nowhere) take on different tasks. It could well be that when you called "your bank" the other bank was originating the call and the person at your bank didn't know how the backend was really wired.
