Wasn't V'Ger damaged when the alien machine planet found it? It must have reconstructed the programming from what was left, and not knowing any better, taking every found detail literally.
If all the aliens saw was V___GER, and the probe couldn't tell them any different, why wouldn't they assume that was the probe's name?
Moreover, V'Ger seems to be totally unaware of the aliens' role in transforming him from a primitive computer to a living machine, even though there is a digital image of their alien homeworld. If V'Ger is aware of those aliens, he seeks humanity more the way, let's say, an adopted child seeks out his biological parents.
It certainly would have been possible to have the crew of the enterprise match the probe via visual identification only and to have V'Ger go by a wholly unknown name. It was more dramatic to have V'Ger seem in some way to be suffering from amnesia or brain-damage, and to only have a fragmented sense of self.
Certainly on subsequent viewings, the mystery of what V'Ger is loses its impact. The film rests more on the philosophical questions it asks, which are really not dependent on the 'reveal' at all.
The problem with something like "Into Darkness" is the 'reveal' becomes too much of the whole picture, so they have less rewatchability. JJ relies too much on mystery to tease the audience into a frenzy enough to flock to theaters.
Khan also had a dramatic reveal in Trek II, but the audience knew going into it who the bad guy was, and therefore the mystery of the bad-guy's identity was not the film's purpose. The suspense comes from the characters slowly figuring it out, and being vulnerable as long as they don't know the score (like the surprise attack).