It's important to know that the existence of non physical systematic differences between male and female is as uncontroversial among scientists in the field as climate change is in that field.
It's easy to be "pro science" when science confirms what you already believe. When it says you're wrong is when you show how pro science you really are!
PS The mental and psychological male/female differences are all statistical with substantial overlap. Much like how men are on average about 5 inches taller than women, but there are many tall women and short men.
That's key. It's perhaps one of the main barriers to exploring this topic. I think it expresses itself mostly unconsciously. It can be terrifying to scrutinize differences and know whatever sub-type you happen to be (man, woman, black, white, etc.) could "come up short" according to some cultural ideal. It's even more terrifying knowing that the measurements and measurers sometimes rig the outcomes of said scrutinizations. I'd hope we'd just try to control and address these terrors rather than stop trying to objectively characterize our differences altogether.
In pair bonding species, sex-type differences are few. In the extreme end, some species can even switch gender depending on the local balance. Some fish even "keep track" of who went female/male last time, as eggs cost more energy to produce. Those that cheat are then shunned.
However we do have evidence that fewer men have reproduced than women. This means men have been under greater selection pressures than women, which would lead to specialization.
I think this is an outcome of the length of time that women are fertile relative to the time required to gestate and ween a child, plus the length of time that a man is virile. This introduces an asymmetry between the number of women who can mate and the number of men who can mate.
Your claim of sexual specialization may stand for more complex organisms which follow the mating pattern above, but I don't think there's necessarily a biological imperative to specialize the sexes, beyond the specialization in their reproductive systems.
Why wouldn't it? Other primate species are very sexually dimorphic, both physically and behaviorally. Why should humans be so fine-tuned that there are no sex differences at all in behavior?
I just think the top level post's claim should be reeled in a little bit to not imply that all species that manifest sex should result in sexual specialization beyond the reproductive system.
Why would anyone believe in this?