On Linux, one user account can’t see another user’s files without expressly granted permission. This is how Plex is implemented on Synology; as a regular user account versus being an all-powerful administrative user.
If I may suggest, while you’re setting this up,
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Make a shared folder (Control Panel - Shared Folders) freestanding instead of using the Synology defaults. Synology defaults have the habit of erasing previously granted permission.
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The procedure to give permission to Plex is included here.