Make sure "Options>General>Importing & Tagging>Store tags in external sidecar files if the file type does not support tagging:" has Video ticked.
Many video files formats do not support saving tags into the files themselves, unlike many audio file types which do support saving tags into the files. I'll let you research if MKV and AVI files support saving tags. Even if they do in some form, MC may not support it for those formats. Test by saving some tags, and checking the media file and the sidecar file.
You can check what tags have been written to any media file from within MC. Just highlight the file, open the tag window, and click on the first line at the top of the tag Action Window, which shows the file type, duration, and size for audio files. See the first image. Then the tags stored in the file are shown in the Action Window. See second image. Sidecar files can be viewed with any text or XML file viewer. I use Notepad++ as it is a little intelligent and easier to use.
Regardless, almost all video file types that MC manages end up with some tags in sidecar files. So in your test, if you moved both the video and its sidecar file, MC would import all your customised tags.
If you have specific tags, perhaps custom tags, that you want saved, then go into Options>Library Folders>Manage Library Fields, then for any field (which can be stored as a tag), make sure that the "Save in file tags (when possible)" checkbox is ticked. Most of the important ones are ticked by default. If you have been keying data into a tag that isn't saved in the file by default, then you can tell MC to write all that data to the files, as below.
To have MC to write the metadata you have entered into the library to your audio (or sidecar) files on the hard disk, select a few files for testing and right click on one, then select "Library Tools>Update Tags (from library)". Check the results. If that does what you want, select portions of your library and run the same process until all your files are updated. You could do the whole library in one go, but it is better to test on a few, then update sections at a time, so if something goes wrong you don't mess up all you files. Use views in MC to manage how you select groups of files for the update.