Yes. Plex publishes a Linux desktop app on the Snap Store, and Plex Media Server has Linux builds too, so both halves of a Plex setup can move.
Plex by Plex · Video · Has a Linux version
Plex supports Linux at both ends. The desktop player is published by Plex on the Snap Store, and
Plex Media Server, the part that holds your films and shows and sends them out, is offered for
Linux alongside Windows, Mac and a long list of network drives.
Your account and library come with you. Watch history, playlists and the way your collection is
organised are stored by the server and your Plex account, so signing in on the new machine brings
everything back.
If the machine you are switching is the one running the server, the move is still fine, but plan
it rather than rushing. Copy your media drive across, install Plex Media Server for Linux, and
point it at the same folders. Plex has its own instructions for carrying the existing library
database over so you do not lose watched marks.
There is also a Plex listing on Flathub. It is put together by the community rather than by Plex,
so the Snap Store version is the one to prefer.