I have an older AirTag, which cannot be seen by my iPhone any more but gives a slight beep whenever it gets shaken. Anyone ever heard of the behavior?
I had a similar problem, the 5 reboots fixed it: https://support.apple.com/en-us/102577
IIRC, this indicates that it's linked to someone else's account, and has not been shared with you. The "beep when moved" feature is to alert people they're being tracked.
For example, I let my mother in law use luggage of mine with an airtag still in it and every time she moved it after the first day or so, it would play a noise.
Mine started doing this and the reason was an Android phone I had laying around with an app that scanned for airtags.
Basically it thought it was being used for stalking (close to an unknown device) and started beeping. Turning off the android phone stopped that.