The 486 couldn't run "any program of your choosing" any more than the Kindle Fire can! If the 486 was running *nix then it couldn't run any MS executables. If it's running DOS/Windows it couldn't run Mac applications. The same holds true for the Kindle Fire. It can run Android apps because it's running Google's OS. If you were to put a version of Ubuntu, it could run a wider variety of Linux programs (probably to include your BBS application). If you can manage to load an embedded version of Windows to the Kindle Fire, you'll be able to run Windows executables. That's an issue of operating system, not the definition of a computer.
But there was nothing stopping him from installing DOS, Windows, OS/2 (back then), *nix, or any other OS of his choice, to run the programs of his choice. Good luck doing that with a Kindle.