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Why can't he just say his son or stepson?
Correct take. "Wife's son" taken genuinely gives off very strong distancing vibes. It feels like he's trying to disown or reject his relationship to the stepson. It's kind of like when people say "baby mama" to refer to their current partner. No, there's no direct link between them and myself: we're just related by this other secondary relationship.
"my wife's _" is such a huge meme that it feels forced to even try reading it genuinely when a more organic way of wording it is available.
IMHO you don't get to call them that until they call you dad, and that doesn't necessarily happen the moment you get married.
My mother in law married someone while my wife was an adult. She would never call him dad, and he would never call her daughter, that doesn't mean they aren't family or aren't close.