I know more about Theravada than the other schools so someone feel free to add to this or correct it
It's basically exactly what QinShi was saying. Budai is considered to be an emanation of Maitreya. Maitreya is a tenth stage bodhisattva, which is very close to being a Buddha. And in particular, since Maitreya is the next Buddha of this world system, he's even more closely associated with being a Buddha, and if I understand correctly, is literally referred to as a Buddha in Chinese literature. So, yes, Budai is basically a Buddha.
Sorry, I'm not sure I'm following. Reactionaries claim Aztec identity to what end?