It appears to be needed for people whom have darkness in their hearts; as being in darkness for too long, appears to strengthen the darkness in one's heart considerably, eventually corroding their very being away. It appears to transform to fit whomever wears it, as was highly noticeable in the case of Xion's forms. ( Which is another topic I'd like to personally discuss; why Xion's black coat was at first the kind females wear, and then it turned into the male one when she turned into Sora? Shouldn't it be the same coat, as it was still Xion and not Sora wearing it? So this brings up another question... maybe the black coat isn't necessarily fitting to the physical form, but rather the inner self-identity of the wearer? The perception of whether they're male or female, as in the case of Xion.)
It's still kinda hard for me to believe, but Mickey needing a black coat pretty much hints at Mickey having darkness in his heart too. XD But it appears pretty much everyone does, except for the princesses of heart. ( Ventus is an exception, as he had his darkness physically removed.) Riku, because his darkness was already too great, and he was going under cover for Ansem the Wise... black coat was meant to throw off the Organization's pursuit of him.
The Ansem one is really tricky. He mentions he's a "servant of the world" in KH II, so this could tie into the self-identity aspect of the black coat... Ansem identified himself basically as an avenger so his black coat changed to that form, but when he lost all his memories when he returned to the realm of darkness, the black coat also returned to it's default form.
No, humanoid Nobodies aren't born with the coat on, as when Xemnas found Roxas and named him, Roxas was wearing his "casual clothes". With the data we've been presented, it appears black coats were most likely created by the scientists of Radiant Garden... probably as a product of their heartless experiments. This would explain how Riku and Mickey got their's, as they were in contact with Ansem the Wise, whom was once part of that research team and possibly kept some black coats for himself.
As for Aqua, I believe she was able to endure the darkness since her "heart is strong" as she says in 0.2. ( Though even then, Aqua comments to Mickey how "the darkness is getting to her" and she was about to give up, but Mickey found her in time to restore her hope. The only thing that goes against that logic though, is when Sora and Riku got trapped in the realm of darkness at the end of KH 2... by that logic they should've succumbed to the darkness after their exhausting fight with Xemnas, and being broken spirited after realizing they might be trapped there forever. I guess it's because Sora and Riku were basing their strength on being there together, had either of them been by themselves there at that point, we might've seen a sad ending of them disappearing... right before Kairi's letter finally reaching them. Food for thought. XD Alternate dark ending, lol.
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