First of all, that word is problematic and you should ask your wife to stop using it. What does she even mean by that? Is she upset because you’re teaching your son to be kind and emotional? Because that’s called being a healthy human being, not being weak.
There’s this toxic idea that boys need to be tough and aggressive and that showing emotion makes them less of men. That’s exactly the kind of thinking that creates men who can’t express their feelings, who have anger issues, and who struggle in relationships. You’re actually doing your son a favor by raising him to be in touch with his emotions.
You and your wife need to get on the same page about parenting because kids are smart and they can sense when their parents disagree about fundamental stuff. If she’s undermining you in front of your son or telling him to “man up” when he cries, that’s going to confuse him and damage his emotional development.
Stand your ground on this one. You’re raising a boy who will become a man someday, and the world has enough emotionally stunted men already. If more fathers raised their sons the way you’re raising yours, we’d probably have a lot less toxic masculinity floating around. Keep doing what you’re doing.