At the risk of being too obvious I am going to provide a simple example (that does involve lady parts) of how my wife and I love my son differently...

My wife...wait for it...BREASTFEEDS my son when he is hungry and in need of comfort!

When my son is hungry or in need of comfort and my wife isn't around I DON'T breastfeed him or snuggle him to my bosom. I give him a snack and I try to make him laugh and giggle because I have no inherent bodily capacity as a male to provide for his needs as my wife so immediately does. I can give him a hug and a kiss and hold him tight but it's objectively different than the breastfeeding thing and how that shapes their relationship.

I'll grant some of your above points that not EVERY aspect of relationality is light years apart between men and women but it's enough of a difference to go well beyond societal norms and constructs.

Also, parenting goes far beyond meeting needs in moments. At bottom, parenting is about helping a child be a person.