Somewhat baffling. Maybe try this. Sorry, it seems somewhat pedantic, but what you are reporting should not be happening. I would start by verifying what image is stored in each track. There is no reason for a new image to be copied to all tracks if you only update 1 track.
In MC, open one of the albums as a view and select the first track. Open the tag editor and and make sure Image shows the correct cover art and that it says Inside File within the image. Step through the tracks and make sure each image in the tag editor says Inside File. Open the directory with another tagging program like Mp3tag and stepping through each track, look at the cover art. Assuming they are all correct, close mp3tag. Then, copy a completely different cover art to the clipboard using your browser and, e.g., Amazon. Then, in MC, select a specific track in your album. Then, do Cover Art - Paste for that one track only. It should change the image in the tag editor but Inside File should disappear and it should have the address on disk that cover art is stored in. The image in the track part of the main view will not change, since that only shows the image for the first file in the album. Then for the selected track, do Library Tools - Update Tags (from library). That should update MC to use the image from the file and tell you now many files it updated. It should only update 1 file. If it updates all files in the album, then that is very unusual. To check, I would then open mp3tag again and look at the directory again, stepping through each track and looking at the cover art. Only the one you selected should have the new cover art. If they all do, then something is really strange. You can also use mp3tag on the other album before and after changing the image for the one track to see if the images in the files are being changed.
Sorry I do not have a clear answer.