Maybe. I think this kid, “Red”, was just a kid with Innoscence who happened across Mana and Allen (the dog). Red is still his own person with his own body. However, I still wouldn’t say it’s impossible. Neah seemed to not question Allen’s appearance when he took over his body other than questioning his age. So, you could interpret that to mean Red really is the original Allen who somehow lost his memories and reverse-aged (or maybe was reborn?). Or it could just be Neah was distracted by the fact that his host body was so young that he failed to consider it isn’t really Allen. Anyway, there’s still not enough to know for sure.
Still wouldn’t be quite the same as Oz/Jack, since Oz only existed as a consciousness created by the Will of the Abyss, and never had a real body of his own, other than his Abyssal form. In the case that Red is the original Allen de-aged, it would still be his body, just his memories that are gone.
Assuming that the Dog is Allen, I'd assume appearance has nothing to do with it. Perhaps Nea thought that Red was Allen and didn't know that Allen died as a Dog.
Could be. Things are still pretty unclear so it’s hard to say what the whole deal with Allen was. It looked as though Neah made his human, adult friend “Allen” the successor, so it’s possible Neah never even knew about the dog. Or maybe Allen was a friend of both Neah and Mana, and Mana just called the dog Allen. And when the dog dies, crazy Mana uses Red as a replacement dog and calls him Allen. Who knows ╮(￣▽￣)╭
This reminded me of a manga called pandora hearts, where the protagonist discovered to be in origin a doll and the corpse he grown up with wasn't in reality his own. So my point is, can this be the same situation?