I have MusicBee 3.5.8883 P, and Windows 11 23H2 64-bit.
Just hitting the "Stop" button while something is playing immediately releases it from Exclusive mode, and other audio sources can be heard. Hitting "Play" starts the music and grabs Exclusive mode again--unless another sound app has already grabbed it in the interim.
Check that you are running the latest version of MB. Also, be sure you aren't hitting the "Pause" button instead of "Stop"--"Pause" will not drop Exclusive mode. Some MB skins do not have a separate "Stop" button, be sure yours does.