That is not the line Oda is following, like, at all
Yamato decided to be Oden long before she was introduced to us.
Oda introduced her as Kaido's daughter because that is a fact. She is his daughter.
The reason Kaido called her his son could have been the New Onigashima Plan, when he wanted her to accept her Oni lineage and become the Shogun of Wano (Kaido would still be the leader ofc)
And if Kaido is calling her son, ofc all his subordinates will also do it. If he called her Garp, Naruto, Lady Gaga or whatever, all of them would follow suit .
Same goes for Luffy and his crew and followers. If Luffy plays along and calls someone Yamao, everyone else will follow.
Of course it is Oda's line.
I know Yamato decided to be Oden long before his introduction, don't even know how the heck you read my post to understand something different.
Yamato was introduced as Kaidou's daughter but everybody treats him as his son. The actuality of his persona is this, not an introduction box with zero real impact in the story. When it comes to
treatment, everybody treats him as a he, not a she; yet people in here are salty enough to discredit this fact (which isn't incompatible with the box) for whatever reason that drives so many guys nuts whenever pronouns are involved (beyond this forum, by the way; does it touch something uncomfortable or what...?).
Kaidou calls him his son because he wants Yamato to take full accountability for his choice. It's made pretty obvious in chapter 1.024: Yamato chose to become Oden and Kaidou accepted it but made it clear that it meant war and wouldn't treat it as just a game nor a simple family feud. Becoming Oden meant the whole package, including his treatment towards Yamato. Before this event Yamato wasn't treated as a man; hence why his epithet shared by the Beast Pirates used to be Oni PRINCESS. The New Onigashima Plan had nothing to do with it, Yamato was treated as a woman and daughter for years.
For the rest, even Ulti treats Yamato as a man in spite of giving zero shits about following suit. And I personally doubt Luffy has any influence there, if every good guy treats Yamato as a he is because they aren't pieces of shit and follow the treatment he prefers for himself.
To summarize, what we have here is that Yamato was the princess (not prince) until he started to call himself Oden, after what the Beasts Pirates did the same following the fact he'd take full accountability for his choice, to the point everybody in-verse treats him as a man. Will Yamato drop the Oden persona and reverse into her old she-self? We will see, but right now the treatment she gets is "he/him", so acting so salty over readers doing the same is, again, fishy as hell.