There appears to be an issue with the Plex authentication server, started maybe an hour ago. My server is running but cannot be contacted via the player.
From this forum initially, which highlighted that I should have checked here as well: https://status.plex.tv/
My server didn’t recover automatically in the end. I had to login to it directly via web browser using it’s IP and port and then claim the server to associate it back with my account. After that everything was ok.