You have the option to use embedded metadata if desired. See Your Media documentation. Specifically the section on Local Media Assets (Movies, TV).
To further confuse things, Plex is in the process of moving to a new movie agent starting with the PMS 1.20 releases (Announcement thread, Additional Details). A new TV agent will follow sometime later.
If you are using the Plex Movie (Legacy) agent (listed simply as Plex Movie in pre-1.20 releases), you’ll see an option for Local Media Assets. This needs to be enabled if you’re using external subtitles, local artwork, etc. It also tells Plex to read metadata embedded in MP4 files. As a result, most people enable it, but drag it to the bottom of the list. This ensures that Plex tries the online matches first, falling back to embedded metadata if no matches are found.
If you are using the new Plex Movie agent in PMS 1.20+, the location to select things has changed. Edit the movie library and choose Advanced. With the Plex Movie Agent, you’ll see checkbox options for Use Local Assets and Prefer Local Metadata. See screenshot below.
FYI, if you’ve a Mac, check out subler, subler.org. It is an excellent Mac only tool for remuxing and tagging MP4/M4V files. It can import metadata from multiple on-line sources. It also makes it easy to edit metadata for personal movies or other videos not listed in online sources. It can also “optimize” mp4 files for better streaming. It rearranges some header bits so clients have to buffer less data before playback begins. You’ll also see it called “web optimized” (Handbrake), “mp4 fast start,” “mp4 fast streaming,” or similar terms in other tools.
Example using new Plex Movie agent:
