UPDATE: gitlab-monitor did not seem to be an issue. I did a comprehensive backup, reinstalled GitLab 13.0.5 from scratch, and restored the data. Now gitlab-monitor was gone and all gitlab services were running ok, but the 500 error was again there. A look into production.log showed an error related to processing Appearance. An identical problem was described by another user in the post Error 500 after upgrade to gitlab 13.0.3 from 12.10.
I eventually gave up and reinstalled GitLab from scratch again, this time without restoring the previous data. The repository was ok, as I could push it back from a local copy, but GitLab metadata was lost. The system is working now, so at least that is good. Now we just need to sign up the users again and repopulate the issue board. Luckily we have only a handful of users and not more than around 200 issues.