Strategy boardgames like chess in the Middle Ages were played primarily by the literate - and hence relatively aristocratic - classes, so it's likely that the prevalence of two-player games reflected the difficulty in organizing strategy games with more than 2 players.

That said, there were a variety of multiplayer games such as Chataruji that had been invented in India by the Middle Ages period; which caused some 19th century historians to (wrongly) trace a link between these games and chess. So people somewhere in the Middle Ages time period were already playing multiplayer strategy games; we just don't have evidence of it happening in Europe.

Finally, I would note that a four-player chess variant was independently invented in Europe by the 1880s, albeit it never caught on and is rarely played even today. It does, however, indicate that somebody in the Middle Ages may have fiddled with a multiplayer chess variant that did not survive to the present due to obscurity of documentation or lack of players.

Edit: For clarity