As the CIA ate my post, I'm gonna say it again (or what I remember I wrote...) for the sake of an argument that also was cut by the CIA. Damn CIA!
Young children are able to recognise sex differences. I've tried to find the actual study, but google fills the top results with gender bullshit about kids and gender. The truth is that young children have a minimal acknowledgement that "mom" person isn't the same as "dad" person and are a bit able to identify another "mom" person. They don't need to understand all the characteristics that are part of what makes a woman a woman, they only can catch some obvious factors.
With that being said, Jazz didn't need to know much to realise that "girl" is something he wasn't, specially if when he was very young, he was given "boy" toys that were different from Ari's "girl" toys. Or "boy" clothes that weren't like "girl" clothes.
For a child's mind, the obvious conclusion could be "if I am a girl, I can have girl toys/clothes" and then, "I want the fairy to make me a girl", as tales with fairies making people other things are common. For any reasonable adult in a more reasonable time, this would be only some anecdotal moment that would end by telling the child "no, you can't just become a girl" and laugh it off. Of course, telling that to a child nowadays would be consider child abuse.
If you ask me, I think that's more likely the situation in that household: Jazz was into girl things (or just went through a phase) and the stupid mother indulged on it, pushed by wacko doctors around her and her own need for fame and attention. The result is the same as treating a girl like a princess: she will grow up spoiled and convinced that the world should treat her as such because she's entitled to. You can "groom" a child to believe whatever you tell them if you raise them in such way. Whatever the family says now is either a misconstruction of what really happened or a lie to make themselves look less abusive.
