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I'm actually starting to think that Yamato her claims of 'being' Oden and Kaido's constant mentions of 'my son' are completely unrelated.

The interactions between Yamato and Luffy this chapter about who was more "Oden" really gives the idea she is just role-playing. Pretending to be Oden gives her the most sense of freedom she can have locked up on this island.

In the mean time, Kaido seems to not care about his daughter's wishes at all. Somebody who locks up their child under penalty of death and plans their entire life out for them does not strike me as the kind of parent who listens to their child personal pronoun wishes.

Just like Big Mom, Kaido doesn't habe a child, he has a tool, a pawn, a chess piece. Kaido wants his chess piece to become shogun of Wano but the problem is, the shogun has to be a male to be recognized by the people of Wano. This is why Momo is the true heir and not his older sister. Kaido needed a son for his plan to work and so he simply started calling his only child "my son" so that others would fall for it.

Though we have to wait 2 weeks to learn more...