Plex thinks two devices are one server

If the second system was a partial clone of another (Time Machine, Migration Assistant) there may be a simpler process. It shouldn’t be necessary to delete the database/metadata/Library config, or other configuration on the server, only the unique Identifiers.

On the newer machine -

Plex not finding libraries after restore from time machine and Big Sur update - #9 by Volts

Then connect, log in, and claim it again, if necessary.