PSA: If you are the owner of your Plex server and follow the _Sign out connected devices after password change- as they suggest, your server claim will also be expired.
So you'll have to get a new claim from https://www.plex.tv/claim and set it on your server; through the PLEX_CLAIM env var if your setup involves Docker.
They talk vaguely about it under _Common Issues_ but it wasn't on the original email, so I lost 15 minutes of my day because of this...