So I'm curious if it's possible to install a Game on my internal Desktop Hard Drive, and have essentially a clone of it on my External hard drive, but they are the same install so when I take my external with me it's the same game files and can be used on another computer, Essentially I want two installs that are the same install rather than just having it only on the external hard drive.
There really isnt a way to do what you are trying, and even if there was its an extremely complex work around to a very simple issue.
Why do you want to have multiple yet single instances of a game? Why not just use cloud saves? Or simply transfer the save file when you need it.