TLDR version: I‘ve had this exact issue since mid May. it got way worse for me, apple found no hardware fault but eventually agreed to refund me.
Full story:
issue started mid May. popup was the same you get when first pairing but with error. Held down crown to re-pair each time it happened (every day when I first used).
I went through a heavy use week and notice every 2-3 hours the APM would disconnect and when I went to my phone, this “not your APM” pop up had returned. Only now the hold Digital Crown to pair was failing. Only way to reconnect was to factory reset the headset.
obviously this became very annoying. Online tech support went through the usuals. Reboot iPhone, no joy. Remove device from Bluetooth and start over, no joy. They eventually bumped to phone support and took in for repair. They arrived back with me two days later with no fault found. Issue persisted.
Phone support refused to issue a refund despite being eligible under U.K. consumer rights act. Bumped to in store support.
Store saw the issue first hand and noted. I agreed to a second repair attempt Which would take “a few days”. 9 days later they were returned to store, I went back yesterday. they run a more thorough diagnostic this time and had tested each component and firmware. No fault found again, likely a software issue. I took out my phone in front of the great support chap in store and the issue just came right back up. After explaining this long story and pointing out my rights, I had a refund authorised.
it seems apple have no fix for this issue. Nor a clue as to the cause.
I know I could’ve asked for a replacement, I really liked everything except the ”bra” with these… but tbh all the hassle and such a weird Software like this didn’t inspire confidence in me.
I hope you have better luck than I did.