He isn't Fai.
That's what everyone calls him and how everyone thinks of him, but it's a borrowed name, just like the life he's living. After all these years his body responds naturally to the lie, turning his head when people call his (brother's) name and smiling while he introduces himself with those three letters. But every single time, a bitter taste remains on his tongue.
He isn't Fai, but he can't say he's Yuui either; Yuui died on that valley too. He keeps on living a life that's not supposed to be his, never letting anyone get too close, hiding (from the truth) behind jokes and laughter and smiles. He is almost happy in Celes, but whenever his fake act starts becoming a bit too real he reminds himself that this happiness is not his to have. He's merely waiting until he can give this life and name back to the one he took it from.
The moment to start his journey is brought on by the most unexpected of reasons, and he leaves Celes with Ashura-ou betrayal's and their last confrontation weighing his heart (is this his fault again?). Like the man who warned him of this journey said, he is joined by three more people, and they set out to travel across dimensions in search of the feathers.
The lies he's so used to telling keep falling from his lips, but it's more difficult this time. The kids are not what he expected - they are kind and loving and care about him (they don't know the truth), and sometimes he catches himself wishing he could be the good person they seem to think he is. And then there's Kurogane, who sees through his act and makes him feel so exposed. It scares him, because he can't afford to be vulnerable when the time comes and he has to kill this man.
He tries his best to stay away from all of them, he really does. But his mask is slipping and he worries about the kids' welfare and he gets an ache in his chest whenever he thinks about Kurogane. He wants this to be his life, to love these people and be loved by them, to forget that he's not really Fai. The name doesn't feel wrong when the kids or Mokona say it. Even if Kurogane never refers to him by name, he does tell him that his past doesn't matter and wouldn't it be great if he could just believe that? (he's so selfish.)
He can never have this, though, because his past is always there, and it finally catches up with him. He's back in Celes and the truth is right there in front of his - of their - eyes. He feels the almost-life he had slipping through his fingers, the guilt and shame weighing on his shoulders, and waits for their hate to turn his way.
Except it doesn't.
Things happen in a blur; he finds out there were more lies in his life than the ones he told himself, but he can't worry about that when the ones he loves (there's no use denying it now) are being hurt. He should live until he can give Fai his life back, but he can't lose these people either, so he unleashes his magic with a mix of desperation and regret.
He's useless against Ashura-ou's power, though, and it's Kurogane who ends it all. Before he can process what's happened, he learns the truth - the full, unvarnished truth - and it's relieving (he's not a murderer) but it also hurts, and as Fai's body fades he can't help the tears that fall (he's so sorry). Fai is free now, but where does that leave him?
It isn't until they are in Nihon that he has time to think about it. His entire life he forced himself to stay away from things he couldn't have - friends, happiness, love. But they found a way into his life anyway, in the form of those kids - clones or not, he loved all of them dearly -, a tiny magical creature, and a man who gave him more than he'd ever imagine possible. They make him feel real, like this life is his, like maybe it's not wrong for him to want to be by their side.
He finds it easier to smile without it having to be fake; his laugh doesn't carry the heavy weight of his lies. The person he is now was forged from the bonds he formed with the people he traveled with, from the sacrifices they made for him and he for them, and from all they went and would still go through together. Little by little, the name Fai stops tasting bitter on his tongue, and even though he will never (can never) forget his twin, it doesn't feel like he's living on something borrowed. This is his own life, and he was never Yuui to the people who are part of it. He's Fai.
The next time he has to introduce himself, it doesn't feel like he's lying anymore.