After updating Plex on Synology, all libraries are offline

Please show me screenshots of the Permissions tab in Control Panel - Shared Folders for the media share you share you are having trouble with.

If you are using the videos share and have Video Station installed, conflict will exist.
It is for this reason we always recommend making your own shares for each media type, named how you want them.

Please do remember, as of 1.15.4.994, user Plex was removed from Administrator privilege.
Please see my announcement here, complete with How-To for assigning permissions to user Plex for your media however you want it.