I had a similar experience with you and FeloGoGo.
How I perceived the problem:
- I paired airpods to my iPhone as usually
- When I open the lid, it would still continue saying "Not your airpods - Connect" even though in background my iPhone still shows Bluetooth settings and there I see my airpods changing status to Connected.
- Close the lid, open it, again
- same story, airpods work (connected) but the annoying "Not your airpods" pop up again and again. Resetting airpods didn't help.
- My Macbook wouldn't show the airpods in the speaker menu (the only way to connect would be to pair it as a generic bluetooth device but that doesn't allow to switch airpods between devices quickly - bummer).
My guess at the root cause:
It turned out, that my Macbook had iCloud semi-connected. I remembered that when I connected iCloud, it asked me to type my AppleID, then AppleID password, then the passcode that unlocks my iPhone. Well, at that point I thought that was unnecessary and creepy so I didn't enter my iPhone's passcode. So that's why my iCloud kinda worked (even Handoff between the iPhone and Macbook) but the airpods still complained. Switching between airpods between macbook and iPhone didn't work (only manually reconnecting as a regular bluetooth device, which is annoying and slow - unacceptable).
I think when Airpods connected to the iPhone, it tried to send info about the Airpods to all other device connected to iCloud.
But since my Macbook was only semi-logged in to iCloud (read above), my iPhone would get an error in response or something like that and then probably a standard reaction to an error is to assume the airpods are not mine.
I'm a developer so I guess Apple wrote try/catch in the code without really looking into the error response and not differentiating on why the error happened, just reacting with the same "Not your airpods" to all errors.
Fix that worked:
- I signed out of my iCloud on my Macbook,
- then signed in again into iCloud on my Macbook with using both AppleID password and my iPhone's passcode.
- Airpods got recognized in the speaker menu of the Macbook, they just started working correctly. If I open the airpods case's lid now, both macbook and iPhone react to it - as expected.
For purity, I'd remove airpods from the iPhone and macbook and reset&pair airpods again but I don't remember if I actually did it. After the iCloud full re-login, airpods started working on both iPhone and macbook and started switching between these two devices (though still manually, you need to choose airpods as your audio like you normally switch between bluetooth speakers and wired headphones - much faster and easier - no need to go to bluetooth settings).