One thing to note is that The Void is not only in multiple Final Fantasy games, but also one of the few things that is ALWAYS the same across games. As well this is how Gilgamesh is able to get everywhere, inculding XIV. Also the (very stronge) possibility that Cloud of Darkness is the exact same one from III. So most likely there isn't an actually duality there, though a connection of course can't be ruled out. Wouldn't be the first time a villian has tried to cast everything back into The Void.

My own personal theory is that Hydaelyn and Zodiark are seperate beings, and not even neccessarily antagonistic to each other...more so since it seems one of Zodiark's "strongest" has his own plans that may actually be detrimental to it. As well...the Zodiark of XII was child sealed away because the Occurians realized he would become more powerful than themselves...maybe here too, he is sealed?