Either one is correct, and they have slightly different shades of meaning. Expanding your sentences to make those meanings explicit:
- He wants to play team deathmatch in the online game, Fortnite.
The preposition in emphasizes the idea that you are metaphorically "inside" the game-world. The preposition in is frequently used this way with video games: "Which character do you usually play in Mario Kart?"
- He wants to play in team deathmatch mode hosted on a Fortnite server.
The preposition on emphasizes the technological infrastructure of a MMOPRG and the fact that you're logging on to a shared deathmatch server along with the other players. You would not use on when talking about a stand-alone console game, because you're not logging on to any remote server.