Apple watch starts playing music automatically

Yes sure. In the Watch app on your iPhone, select Workout and then under Workout Playlist, make sure None is ticked.

That should fix it.

Is that during workouts - or even without? Just asking because workouts can trigger music automatically. Or is there any other pattern to that automatic playback?

This should normally not happen, regardless of that auto launch setting.

If the issue persists, I would suggest unpairing and re-pairing with your iPhone. That will return your watch back to factory settings, while keeping your apps and data intact. Start from here:

Unpair and erase your Apple Watch - Apple Support