The solution was fairly simple but tricky to find with experimenting.
At first I found out, that spotify has a option "Start Spotify on Login". I deactivated it and restarted Spotify manually. And suprisingly: Spotify started on the second screen as wished.
But the function that Spotify starts on Login is convenient. So I put a shortcut of Spotify.exe manually in the windows autostart folder:
C:\Users\[User]\AppData\Roaming\Spotify\Spotify.exe
in
C:\Users\[User]\AppData\Roaming\Microsoft\Windows\Start Menu\Programs\Startup