Plex Media Server is running on the Synology NAS, correct?
Are your movies/shows/etc located in the /Plex shared folder?
If yes to both questions, you need to move your media to a new location.
Locating media in /Plex will not cause the issues you have with the Firestick.
However, it will cause other problems down the road.
Suggest you read the two support documents linked below. They have information regarding media organization and other useful information when running Plex on a Synology. It will save you some future headaches.
