English would work for titles:
He is President.
He is Doctor. (Dr.)
"He is president" is different than "he is a president" (one among many) or "he is the president" (the one specific).
But it would have to be within the context of titles
With the specific case, the "the" could be dropped - "who is in running this rpg?" points "he is gamesmaster" which means "he is the gamesmaster".
Probably because it's a role, similar to the title?
That sounds odd. Like in your example the nature answer would be “He’s the games master”. Without “the” article, one could get confused like okay is he the game master for this game or is he just a game master for some other game
I think because someone wouldn't answer "he is a gamesmaster" to the example question, "who is running this rpg", it makes it less odd.
If the Q was "who can run this game?" maybe then it would be ambiguous because it could imply others may also be a gamesmaster.
So the context of dropping the "the" can only really happen when there's no ambiguity.