The Iliad and Odyssey were probably composed in the first quarter of the 7th century BC. At that time it was apparently considered plausible that Odysseus' domain could muster 12 ships with a crew of 50 each (for a total of 600 fighting men).

Now, firstly, it should be mentioned that this is one of the smallest contingents in the Catalogue of Ships. Most regions are said to have sent at least 30 ships, with some sending as many as 80 and Agamemnon bringing 100 of his own. Only Rhodos and the tiny island of Syme sent fewer ships than Odysseus. Apparently the lands he ruled were considered some of the most sparsely populated (or poorest) in the Greek world.

Secondly, Odysseus' 12 ships were not crewed purely by men from Ithaka. In fact, the poem declares that Odysseus' men were Kephallenians (not Ithakans), who inhabited Ithaka, Neriton, Krokyleia, Aigilips, Zakynthos, Samos, and the shores of the mainland opposite. The island of Kephallenia/Kephalonia, the largest in the area, is conspicuously absent from this list, despite giving its name to the followers of Odysseus. Some scholars have therefore thought that the "Ithaka" of Odysseus should actually be sought on Kephallenia itself. Either way, his domain included far more than just the island we know as Ithaka. Since both Kephallenia and Zakynthos are considerably larger than Ithaka, the poet probably assumed only a small minority of Odysseus' forces (perhaps no more than a single crew of 50) actually came from Ithaka. Even a tiny community would be able to support such a small expeditionary force.

Indeed, it seems Ithaka always remained too small to be of any significance in later history. None of the Greek historians of the Classical period make any mention of the island, despite the fact that the Athenians were involved in numerous operations on the larger Ionian islands (Zakynthos, Kephallenia, Leukas, and especially Kerkyra/Corfu further north) during the Peloponnesian War. A place as tiny as Ithaka would not have been worth any specific mention or effort; as a community, it may simply have been considered part of the bargain upon the subjection of neighbouring Kephallenia.

This is not to say, though, that the island was uninhabited. Later sources report the presence of a "city of the Ithakans" called Alalkomenai. This town seems to have existed throughout the Hellenistic and Roman period and its archaeological remains can be visited today. I'd have to check the Inventory of Archaic and Classical Poleis to see if we know anything about when Alalkomenai was founded, but its presence suggests continuous habitation of Ithaka in historical times. It was simply too small to draw any attention to itself.

If the island is uninhabited in the game, then, it is a decision by its designers, with no basis in historical fact. The game is keen to include elements of the mythical past within the Greek world of the Peloponnesian War, but it seems to try to keep the two somewhat separate. You have to hike away from the historical world to wander into the mythical one. This is what allows them the artistic license to even have a "palace of Odysseus", of which no actual trace exists.