Hi jwilander,
I did reinstall everything from scratch and in fact realised, that when I start mattermost with systemctl the service wouldn't run.
The reason was, when I first executed platform, I did so as root. So the log file was created and only writable with root rights. Systemd though starts the mattermost.service with the user mattermost and thus mattermost couldn't write into the log file and exited.
Thanks for your help jwilander