Well, he is as old as the Fallout Universe itself.
Either he is an incredibly well preserved Pre-War Ghoul or something FAR more interesting.
Think about it:
-He is seen everywhere in the US.
-He has been reported many, many times over hundreds of years.
-He only shows up long enough to make one single, seemingly random, kill.
-He disappears again just as fast.
Baring 'A Wizard Did It' there is only one real explanation for this:
The Mysterious Stranger is NOT one man, but rather a Faction of some kind, even more secret than the Institute. One that sends lone operatives out into the world to take one single decisive action for a reasons that are beyond our understanding. Operatives that are never seen until it is too late, who never fail, and are never defeated. And they seem to have a vested interest in keeping the various Heroes Of The Wasteland alive.
So, who could this Mysterious Faction be?
They must be even more advanced and shadowy than the Institute, yet they do not seem to involve themselves in the World beyond saving the various Heroes.
They must have access to superbly trained and equipped personnel.
They must have SOME way to predict where and when they are needed.
If we limit our search to 1950's era organisations, two jump out:
D.A.R.P.A. :https://en.wikipedia.org/wiki/DARPA
They would be the source of the most advanced technology in the world (barring the next one) and have access to the finest personnel in the world (barring the next one). They could have perfected the technology that the Railroad uses with P.A.M. allowing them to identify individuals who will play an important part in the reestablishment of human civilisation and provide them the protection they need when they need it. They could be motivated by altruism or, more likely, their predictive models show that any overt action on their part would end it total disaster.
The Manhattan Project: https://en.wikipedia.org/wiki/Manhattan_Project%C2%A0
The finest collection of the finest minds the world has ever seen. Somehow, the very Founders of all that is Fallout are never mentioned..........strange, that. Perhaps the Project never ended, researching ever more powerful and terrible weapons of war until the inevitable, bitter end. The resources they would have access to would be unimaginable. And so would the GUILT. Picture it, a Faction of near unimaginable power, so traumatised by past abuses of that power that they are nearly paralyzed. Only intervening when the fate of the world rests in the balance, and then by doing as little as possible. It would be Issic Assmoves' First Foundation.