Presumably, all the "aether" in the 13th Shard/Void is contained in "living" beings. We do already know that the people of the 13th Shard played around a lot with Auracite of some kind and that this ultimately corrupted them and lead to the Flood of Darkness. One of the big things we see Auracite doing in general (and regardless of type of Auracite) is being really good at absorbing aether.

The thing is... "living" beings in FFXIV are made out of aether physically, the same as anything else. And we know the Voidsent aren't physical in nature. Not really anyway. Anytime we see Voidsent on the Source, it is not as they are in the Void. The Voidsent on the Source are possessing bodies that physically exist on the Source. When Voidsent are "killed" on the Source, they don't really die either. Instead they get severed from the body they are possessing and get punted back to the Void. This leads me to think that that "aether" the Voidsent have isn't physcial aether (which is what most elemental aether is) but the kind of aether that makes up souls/ideas/memories. That is, aether that is more associated with "mental" concepts rather then "physical" concepts.

It's that "physcial" aether that is completely missing in the Void and it is also that aether that is what the Empty is comprised of.