Ha, there actually
are engines, that many of the new RPGs share. But I am not sure D&D (or at least 5e) is well suited for being a base for many different games, because i.m.o. it has a strong focus on combat.
If you want a game with lot of encounters with monsters to defeat, getting more and more powerful to defeat even harder enemies, with combat system that is not overly complex but still tactical, you definitely could (and maybe even should) base your game on D&D
Heroic fantasy works. I could imagine Witcher, or Monster Hunter, even aliens style sci-fi reskin, fighting monsters in derelict space-ships should work... but i.e. Shadow of the Colossus would be harder, and Game of Thrones style political intrigue might turn into really simplistic dice-rolls.
But this is probably true of many games, i.e. I heard that Fate is really good if you want to do something with a structure of action-adventure tv-show, but probably not a psychological horror :P But doing i.e. politics in that system might be more fun than in i.e. base 5e D&D.
Probably the most succesful engine I have seen is in the powered-by-the-apocalypse games. Apocalypse World invented a really simple base machanic, where if your character wants to accomplish something, you roll two 6-sided die, sum the numbers, and if it is 10/11/12, you just do the thing, if it is 7/8/9, you do the thing, but something bad happens, and if it is 1 to 6, you don't do the thing and something bad happens.
Probably my favourite system I have played so far :-)