Synology running Plex will not even recognize .mkv files in my TV directory

That is only one part of what is asked for. The paths and Plex dance will also provide more information. It could be a permissions issue or something else. Which is why we ask for all 3 pieces of information. (the file names and path give us information that might be root cause, or just what to look for in the server logs) The server logs then can give us insight on what Plex is seeing/doing