I updated to iOS 16 and that seemed to solve all the problems I have. Specifically, in iOS 15:
- The AirPod is not listed as a device in "Find" on iOS and macOS
- "Days left to purchase AppleCare" is stuck at 61 days
- The AirPod does not have the correct icon / shape in menu and settings, both iOS and macOS
- Other small, weird things
So, the minimum iOS version needed for AirPods Pro 2 (full functionality) is iOS 16.