Like the in-game passage of time not actually being reflective of the actual storyline time period, the use of game servers requires a suspension of belief - narrative wise, there is no such thing as a 'server' - everyone is playing together in the same world from the point of view of the player. Yes gameplay wise your character Burning Passion is on Balmung and my Enki on Ridill, but there's no difference from the story's perspective - we're basically on the same world along with every other player of the game.

As for the Void, that's used as a catch-all term by Eorzeans to describe anything 'not of Eorzea' - hence the wide variety of monsters called 'voidsent', ranging from demons to squirrels of all things.