Apple Music adds music to playlist which isn't really added to the list
Well I just tested the other way round. Sorry I don't came on this on my own as I posted the question. 

I edited the playlist instead, search the currently playing song, added it and saved the playlist. This worked for me. But during the save process the iPhone really stutters heavy. It nearly freeze for a few seconds. This can be caused by the temperature currently in Germany (38 degree) but it's still strange.

Anyway, it should still work the other way around. But if someone else encounters a behavior which is quite the same, he can try to solve it like I did.