@matthias.graehn Just to confirm: you are receiving an email to your matthias.graehn@edu-werkstatt.de account after the test email command, right? If you are, the notifications from the app should deliver okay.
The part you quoted above will show even if the mail server does not deliver the message for another reason - it might accept the message, but won’t deliver it. It might get quarantined, for example.
If your domain does not require smtp-relay then I would not use it.
You can also turn on debug temporarily as shown here. Maybe that will produce something. Check your /var/www/passbolt/logs folder for a debug.log file. If it’s not there you can create and grant permission to the web server.