Very unlikely this is a bug with MB as the forum would have lit up with reports.
I don't have another keyboard, but I've tried the rest and it's still broken.
Interestingly, it breaks from the second+ time I press "Play/Pause":
-If I press it once before playing any track (after re-opening MusicBee), then it will be broken if I start a track (by mouse clicking) and then re-press "Play/Pause".
-If I press it once after playing any track (after re-opening MusicBee), then it will pause the track correctly on the first click, and break from the second+ (it will quickly pause and play the track again, in like 0.1s)
This heavily hints that it's a problem with MusicBee.
Win 11 Build 22523.