It's indeed not running as root and I was also testing it as root before I made it run under a new user.
I changed the ownership as instructed, but it didn't work. Still getting shm_open() failed: Permission denied.
I also deleted the teamspeak/musicbot related files in /tmp/, because some had root ownership and tried again. without success.
It always seems to be the first instance that is being started, that fails to connect, but reports that it was running and that cannot be shutdown. All following instances work correctly:
With DisableLocalXServer set to true, I managed to remove the invalid running state. When I then start the server with DisableLocalXServer set to false and choose to start the second instance first, then this instance will make the trouble and the original instance will work just fine. I can then remove the second instance and add it again: Those instances will work just fine as well. If I started the first instance first, I couldn't remove it, but use the second instances just fine.