I believe the world of Hydaelyn exists in it's own universe, or at least it's own galaxy. The same can be said for all the other FF games, except perhaps FFXIII /FFXV (same mythos) and FFVII /FFX (Gaia's ancients being X-2 Shinra's descendants).

It's then likely that all of these different universes/galaxies are connected by the power of the Void to form a Multiverse. It's that null state that forms a (probably temporary) bridge for characters like Gilgamesh, Lightning and Shantotto.

e.g. If this diagram is an analogy of the FF Multiverse. The void would be the space/gaps in between each "universe bubble":



It's possible that what is referenced as the Void in FF XIV is actually a subset of this FF series void, or perhaps even another kind of dimension/world that exists separately all together.

In any case, I get the feeling that FFXIV will explore it's own interpretation of what the void means to Eorzeans a much greater deal as the story expands. (FF stories tend to always progress towards a cosmological/metaphysical understanding as they become developed)