To start off before my theory I want to explain some things that this theory isn't.
1. Not a theory that XVI is one of the shards.
2. Not a theory that XVI is on the same planet as XIV in the past, present or the future.
3. Not a theory that XVI is the same, but an alternate version of the reality in any form. Though I do believe that the universes are adjacent, just not in the same way that XIV has multiple shards. Its not a single planet kind of adjacent, but a whole universe being adjacent to another.
Now, to further set the ground we need to explain the Interdimensional Rift, and how it is known to work. I am not including crossovers from other games unless it relates to FFXIV.
1. You can dream your consciousness across the rift if you have some sort of an anchor. I.e. characters being pulled to sleep, and waking up in Dissidia NT. I know the game wasn't a very good one, but it is a perfect example of how they can tie characters into crossovers.
2. You can suddenly feel "lost" and not know where you are, but in reality be in another Final Fantasy game. A great example of this is Iroha from The Maidens Rhapsody event that sometimes comes back in FFXIV.
3. You can somehow be transported to the world of XIV, and be completely aware of it in the case of Lightnings FFXIII event, which makes sense. There was a lot of worldly bending stuff going on in her trilogy, so she would possibly notice if she were somehow conscious in another one.
4. Shantotto is just magically involved in a fate during her XI event. It is not explained how she got here, but she is here. She isn't a strong example, because we don't know how she was in XIV, but I feel the need to at least put it here.
5. The Puff Puff girls, and the monsters from Dragon Quest somehow making it into the world of XIV. Likely leaking into the game in some way, which the quests hinted at.
6. Yokai Watch monsters somehow bleeding into the realm from a completely different universe made by another gaming company. Square Enix does not publish those games. This one is the least likely to be canon, but considering the interdimensional rift doesn't just work for Final Fantasy games. It seems to tie in a lot of Square Enix titles at times.
7. Noctis literally followed a daemon into the world of XIV, because said monster could travel the interdimensional rift. This is not the first time the actual Noctis from XV has traveled to another world either. Dissidea though weird, IS canon, but it happens in a way that isn't necessary to play. Play it to make sense of it. Though, I would suggest playing an older one. The older ones are a lot more fun to play, and don't require online services that are discontinued.
8. Gilgamesh often travels through the Interdimensional Rift. Ever since his first game in FFV he has been mostly the same character every single time, bringing weapons from past games on his travels with him. The only debatable different Gilgamesh is the one in FFXV.
9. The Omega Weapon. Though XIVs MIGHT be different, it is theorized that they might all come from XIV. Either way it DOES travel dimensions, and is known to come from other Final Fantasy games.
10. Shinryu, in its original incarnations it is literally FROM the Interdimensional Rift itself. Though the XIV one is a mere imitation, it is possible that the original was a traveler, and that it is based on that one. Though not much is known about that, and its merely speculative. Take my theory on that with a grain of salt.
11. Y'jhimei is a Miqo'te who traveled from the universe of XIV to XV's. In her lore she was literally transported from XIV to the world of Eos in XV. It is canon that she traveled to it. A weird case, but it happened.
Now are these events all completely canon? OF COURSE NOT, but the Final Fantasy ones are definitely canon. The events that place in the Final Fantasy series CLEARLY describe inter-dimensional travel, and how it is possible. The Dragon Quest one is iffy, but considering they own the rights, I don't see why not. The Yokai one also might be canon, though that is the weakest of them all, they do mention "coming from another world" during the content. These all imply the rift is involved.
So what makes some characters the same, but not quite in a strange Final Fantasy fashion. This is the next part we have to explain to get to the bottom of this mystery. In the series, there are characters that share similar traits, names, and roles in each story. They don't always come back every time, but usually Cid, Biggs, Wedge, and Princess Sarah often come back in some form. Princess Sarah is on a hiatus, but for awhile she was a reoccurring princess for a bit. Appearing once in Final Fantasy III, and again in IX, though differing people, they are a reoccurring character. This category of character seems to spread like minded copies in the series in some way that pop up throughout the worlds in the series. No one knows why, but they happen.
Now its going to get weirder. Why do places like Ivalice, Doma, and characters like Fran, Golbez, Princess Ashe, etc. pop up in FFXIV, but not other games except for their origins? Well, I think somehow "copies" in a lore sense exist for no reason other than they are copies. Not the same, but eerily similar, just played out in different ways. Somehow the universes can take these characters, and "clone" them from some kind of influence in a way. No one knows how, and I don't know if the devs will ever explain this. They just seem to appear for fan service, but its fun to think of a lore reason for why they are "copied" in a sense. Don't think too hard on it though.
So this brings me to how I feel XIV, and XVI are connected.
I believe they are ADJACENT universes, which CAN influence each other. Another possibility is that they are in the same universe, but are far far away from each other within the Sea of Stars. Though, the adjacent theory in my opinion is more likely, either one COULD work. I think the devs just want to distance XVI from XIV as much as possible, but since XVI is theorized to be an upgraded version of XIVs engine, and they do share similar motifs in style, speech, story, and dialogue, it does make sense that they would be "loosely" connected. Meaning that they are siblings which influence each other in concepts. Though, I think its without a doubt obvious that XVIs universe is darker, and less merciful. They also seem to share a similar concept to how transforming into primals can work. Though I'm unsure if using these Eikons as they are called disrupts whatever makes the world work in that universe/world. We don't know enough about it, other than that the game heavily borrows from the way that Shiva, Thordan, Knights of the Round, Tsukyuyomi, Hydaelyn, Zodiark, Save the Queen, and etc were used in XIV. I feel like XVI is expanding upon this concept, because of its popularity, and interest in the devs. This is one way that XIV influences XVIs world in a sense. I theorize they are SIMILAR, but not quite the same. Siblings if we want to call them. Therefore I will dub my theory the "Sibling Universe" theory. It just sounds pretty cool. :P
Note: Nier DID NOT travel the interdimensional rift to get to XIV, and make all the copies that exist on the first. Somehow Nier got to the First, and traveled after a very long time during one of its endings. That means that one of the parallel endings IS in the XIV universe. Does that mean every single ending is? No. Look at the MCU, and how realities work when they are adjacent. Events play out different, so even though it is true that Nier could be in the universe that XIV is in, it doesn't mean it is. A lot of games like to use multiverse theory, and Final Fantasy has hinted at it before. It is no exception, therefore I think just one ending is linked to XIV.
another note:
I feel like I should note that XII, XV, and Type 0 exist on a different type of universe than the rest of Final Fantasy. They call these Fabula Nova Crystalis, and for some reason share differing god like entities, and methods of universal laws compared to the other Final Fantasy games beyond its rifts. Though they are still Final Fantasy, they share similar universal laws in some way, and represent the idea that in Final Fantasy there are likely sibling universes. Though these don't write the rules for them all, it is a clear example that XIV, and XVI could be similar in a way. As well as Tactics, which I will discuss a little bit further.
Some people even feel that the Ivalice universes also exist in a way, so they try to explain how they all could be connected, and reflected to another, because its not quite clear. Are they just legends, or are they events that all canonically happened? People got very confused for this reason. Regardless though, Square Enix calls XII, XII: Revenant Wings, Tactics: War of the Lions, Tactics: A2 Grimoire of the Rift, Tactics Advance, Vagrant Story, and more obscure titles part of the Ivalice Alliance. They are evidence that some sort of realities exist with similar laws, and therefore they are also adjacent at the very least in some form.
Now this theory is popular, and merely hinted at in X-2, but it is implied that FFX's planet was populated in the far future of VII's planet. Therefore they MIGHT exist in the same universe, and not be in a different universe at all. This part of my post is getting a bit weird, so no I am not going to elaborate any more on these, aside from the fact that I will say that the time of X-2 was very strange for Square Enix, where they were doing a lot different sequels. Some of them didn't pan out well, and were considered messy by fans in ways. It was when the creator of Final Fantasy left that all of this started to happen. Keep that in mind, because when games in that era have references like these, and suggestions that they are loosely connected, it might not mean anything now if they were to go back and remake them. Just food for thought.
Anyway I think that is the end of my post for now. Please understand that I am merely peacefully discussing evidence that FFXVI could be RELATED loosely in some way, and that I don't think they are tied together in an extreme way. I think there will be references (like Estiniens voice actor being a minor boss) at best most of the time, especially because it seems like the devs really don't want us to think its XIV. So much so, I theorize this may be one factor in why we aren't getting an XVI event close to its launch in FFXIV. I believe they really want to do their own thing, and drive home this fact before doing the crossover event that Yoshida has hinted that they will do in the future. Regardless though, I think they are loosely connected, and that is okay. They just won't be focusing on any hint of that for now, until fans realize how different the games actually are during the launch period, but keep your eyes peeled, because you may find Clive, and possibly even another character from that universe drifting into the world of XIV for a moment. ;3