Additional features, settings, or issues not covered by specific Microsoft Teams categories
There seems to be two problems going on with the AirPods and MS Teams: first, the AirPods want to connect with other BT devices all the time (even opening the case will disconnect them), and second, as others have noted, MS Teams seems to confuse them as both Headphones and Headsets.
I managed to get my AirPods Pro to work with MS Teams on a laptop PC by taking two steps: 1) disable all nearby BT devices (phone, watch, other PC's, etc.) and 2) disable the Headphones device under Manage Sound Devices settings when you are connected.
And don't open the case or it will disconnect you! Clearly this is less than ideal, but from this experience I think this is more of an Apple problem than a MS problem.