I'm not sure if this point has been covered yet, so I apologize in advance if it has. I'll start off with a fun movie quote:
"Now, a staple of the superhero mythology is there's the superhero and there's the alter ego. Batman is actually Bruce Wayne, Spider-Man is actually Peter Parker. When that character wakes up in the morning, he's Peter Parker. He has to put on a costume to become Spider-Man. And it is in that characteristic Superman stands alone. Superman didn't become Superman. Superman was born Superman. When Superman wakes up in the morning, he's Superman. His alter ego is Clark Kent. His outfit with the big red "S", that's the blanket he was wrapped in as a baby when the Kents found him. Those are his clothes. What Kent wears – the glasses, the business suit – that's the costume. That's the costume Superman wears to blend in with us. Clark Kent is how Superman views us. And what are the characteristics of Clark Kent? He's weak, he's unsure of himself, he's a coward. Clark Kent is Superman's critique on the whole human race."
So far, we've been talking about the social identity being the "true name", and vigilante identity being the costumed persona.
But what if we flipped that. What if the true identity is the hero, and the social identity ... is just some schmuck the hero pretends to be when he wants to hide out and/or blend in with the masses? I realize this is somewhat contrary to the wording of the Dual Identity class feature, but I think it works much better this way. It also makes a lot more sense.
For example, it is the social identity that gets the +20 circumstance bonus to Disguise checks, not the vigilante identity. In their social identity, the Vigilante could even disguise themselves as another race or gender or age group. What's more, they would probably be able to pull it off reliably and often with that sizable bonus!
But the vigilante identity gets no particular boost to the disguise skill. A human Vigilante would never be able to pass themselves off as a vigilante identity of another race, not without additional magics such as Disguise Self. It is the vigilante identity, therefore, that is most likely to be the true self, w/ the social identity being the costumed persona.
tl;dr: The Vigilante class isn't Bruce Wayne pretending to be the Batman. It's Superman pretending to be Clark Kent.