More so on recent point as to whom the DT spirit belongs to, I would again lean on Nero: Vergil doesn't seem likely volunteer to be a puppet for someone elses gain ^_^.
Here's my suggestion. It's Yamato.
For one, a majority of Devil Arms are Devils before they're Arms. Secondly, a Devil Arm that holds a devil's soul is 100% of the time named directly after the devil it came from. Yamato has had its name and existed since Sparda's time, and Agnus calls it an "extremely powerful Devil's Arm", as in "an Arm that belongs to a Devil", grammatically speaking. The fighting style exclusive to this Arm is described as that of a "warrior fallen from grace" and it has the “will and the power to divide and wipe out the darkness”, and has the ability to destroy or form Hellgates. It'd be pretty screwy if this super powerful Arm was just an empty sword with no soul in it whatsoever. How's it so powerful, then?
Chances are if Vergil had a DA, he wouldn't want to share it with something that has its own name. His own Arm would be named "Vergil" like Sparda's is named Sparda, and it'd have whatever powers he had. But he doesn't have one, and the Yamato can't be him either, because if he could form and destroy Hellgates at will, DMC3 wouldn't have happened.
Surprisingly, that proof is with the DMC4SE. Vergil casts a demonic shadow for a form he's never shown having at all in 3, even though the Fortuna visit is before 3, but the shadow resembles Nero's DT form, and Nero can't pre-natally project himself-- or at least, I sure as hell hope not. Vergil's DT in 3 lacks most of the traits of the shadow he's projecting. It even has a different number of fingers on its left hand, where Vergil has all his left fingers in 3DT but with an extra protrusion to hold his sword that has "two/three claws".
Throughout DMC3, Dante’s and Vergil’s Trigger forms differ between each other depending on which weapon is equipped at any given point. They may or may not have wings, their “hair” is pointed or curved, they could or couldn't have gauntlets, spikes, etc. This continues into DMC1, which is still canon (hopefully), so unless they retcon the way Devil Triggers worked in the first game, then even at the age of 28 or so, when Dante's reached physical maturity, his own Devil Trigger form changes depending on the weapon he has. Alastor looks vastly different from Ifrit, and when Dante equips the Sparda, he looks like his father. This is the only time he actually looks like his dad in DT form, because their forms usually look as different as night and day unless we compare his DDT in 2 with Sparda's form or the Sparda DT in 1. So basically, it takes a good long while post-achievement of maturity for a son's DT to look like a variation of his father's.
So theory is, depending on how strong the spirit inside the Devil Arm is, or depending on how much power the Devil Arm wants to give to its wielder (or how much the wielder needs from the Arm), the wielder could adopt their form and become them, or have his "natural" form with additions from them. The Force Edge even turned Arkham into Sparda. And he's not even related to the Spardas, so at least one of these things I said is true for his situation.
DMC4 proper is where things get screwy, with the game's insistence that “The Angelos are made from fragments of [Nelo's] Armor”, when neither the Alto nor the Bianco actually look like Nelo. Like. Not even close. Even the Savior (supposedly a “statue of Sparda”) has more in common with the DMC4 Angelos (both common enemies and Angelos Credo and Agnus) than Nelo does, and the Savior was constructed from "two millenia of demonic energy". They certainly wouldn't get the two millenia from Vergil. If anything were made directly from Vergil's/Nelo Angelo's corpse/armor, it would have Vergil's features on them, not have more in common with each other than they do him, outside of the one shadow.
Breaking this down:
Nero's Perfect DT form and DT Specter have the same shoulder protrusions as Vergil's Shadow, but Vergil's actual DT in 3 does not have either.
Nero's "horns" in his Perfect DT and Specter sprout from the back of his head. The Savior's does, the Biancos and Altos do, Vergil's does not, and Nelo's come from the front of his head and project further out from his face.
Nero's PDT and Specter have a partially obscured face like Credo and Agnus, where Nelo Angelo's face is "exposed" and delineates human features, and Vergil's DT face is only obscured by his "coat collar".
Nero's PDT and Specter have "shoulder ridges" like Vergil’s Shadow and the Savior, and the Angelo have ridged surfaces. While Vergil's DT does also have ridges, it doesn't interrupt his silhouette in any way and may as well be smooth, even though he visually looks incredibly scaly from head to toe.
Nero's PDT and Specter has wings like all Angelos except Nelo.
The only thing Nero's PDT and Specter, and Vergil's shadow and DT have in common is the left arm sheath+claw combo. But tellingly, both wielders are around the same physical age when they display that trait. From what we see in DMCs 1, 2, and 3, and even 4, it's incredibly rare that father and son share physical features in their DT outside of extenuating circumstances such as actually wielding the sword with their soul in it or-- in Dante's case-- being almost 50 years old and near-death. That arm-sheath can be considered Yamato-exclusive.
Another difference is that Vergil's Yamato DT keeps his toes intact, and Vergil is a half-demon. If Nero's DT were to resemble Vergil's because of a father and son connection, Nero wouldn't (eventually) be missing his toes in that form. He's more human than devil and he probably was born with all his toes. Such drastic change in physical features comes from a strong demonic presence in the Devil Arm. Again, we can refer to DMC1. Dante's Alastor DT has wings, spiked back hair, and IIRC he even gains a third eye as well as what looks like a dragon's head on his shoulder. Dante's Ifrit DT has none of that and he has a giant single eye in his face. I doubt Dante has ever been a cyclops even once in his natural life.
Back to the point: Vergil and Nelo don't have resemblance to the Angelos, the Savior, or Nero's DT. Being that it was Yamato and not Vergil that we can actually see was experimented on to form the Angelo armors, and seeing that no one ever says who that voice belonged to that echoed in Nero’s head, here's the radical idea I'm sideways suggesting:
Yamato exists as its own being with its own values, and those values can be resonated with to access its power. Again, no one says who that voice belonged to that echoed in Nero's head. The man that Nero is described as seeing in his dying vision before he DTs is.... not actually described as Vergil. Really. Go back and read that sh!t. CaT herself says it's "obviously Vergil", but that's her opinion, not actual fact within the novel. The only thing described of the "man" is that his glare is "cold" with a touch of gentleness/kindness, and that this being says he desires "more power" when he's asked. Nero's also described as never having met the man, but with how much Nero and Vergil are supposed to resemble each other, and Vergil being Dante's effing twin, Nero should have been able to note that this man "looked like Dante" or "looked like him (Nero)", or at least had a single mention of white hair or a blue coat.
The unnamed figure also spoke to Nero in his mind/vision/dreamscape/what have you. Alastor and Ifrit have done the same to Dante. Alastor, Ifrit, and most other Devil Arms (especially Sparda) are seen to grant their form and power to those most worthy of it after judging them in some way. Nero resonated with this figure in the most passive way ever. "What does your soul want?" "I dunno, what's yours saying?" "I want more power". "Great, I'll have what you're having". It's different than Al's and Ifrit's "Try to kill the wielder", or Sparda's "has to be feeling righteous rage" requirement (or in 3, "swing my sword a bunch of times"), but it counts as resonance, I suppose. I'd even say that Nero resonated with the Yamato more than even Vergil did. Maybe Vergil was stubborn. Point is, Nero's PDT and Specter have a ‘combination’ of the parts the fragmented Yamato lent to different enemies, whereas the Yamato only gave Vergil a single thing in common with Nero. Yamato would've probably resonated with Nero regardless of blood relation. Blood does not guarantee an identical Devil Trigger form to another, not even on identical twins (see: Dante and Vergil). Lack of blood doesn't stop someone from looking like the spitting image of the Devil in the Arm (see: Arkham).
DMC5 might change all that, but until then, this is what makes the most sense to me. And they'd have to do a hell of a job explaining how "Vergil" in Nero's vision is "cold, but gentle" when he never met the guy, and the man actually smiles before granting Nero his power, whereas "V" in 5 just slices off Nero's arm. If Vergil and V are (probably) the same person and Vergil was also inside Yamato, those actions are too different. It's not like Vergil was in a better state in 4 than he is in 5: the DA he's supposedly inhabiting was broken in half. He should've been even meaner in Nero's vision than he was actually described.
This whole theory may look convoluted judging by the sheer number of paragraphs it took me to explain the reasoning, compare and contrast and all, but ultimately it leads to a straightforward answer as to what could be going on in DMCV.
"V" wants the Yamato because it's OP as f#ck. "V" cuts off Nero's arm because the power to open and close Hellgates is something he wants to actually do or he wants to have it simply to have it. "V" wants the Yamato because he's upset on some level that it granted Nero more of itself than it would let him have, and the implication that he's not worthy of its full potential is extremely insulting to him. And let's be real, Vergil was not using the Yamato's full potential in 3.
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