“Men and women are more alike than they are different.” True, we are both human. More alike is not completely alike. Your use of the word more acknowledges that. Socialization is a factor. The fact that gendered characteristics vary with societies in different locals that you point to shows that. I do think that there are differences beyond our plumbing and other physical characteristics but they need not be considered shackles.
Looking at my sloppy diagram of distributions, for the sake of discussion the peaks/mean at the top of the diagram are equal. Let that represent an amount of inherent difference (nature). The distributions of A and B are different, representing socialization (nurture) where the curves at A are more alike (more in the overlap) and at B less alike (less in the overlap). The amount of socialization required to achieve more alike (A) is be greater if the natural difference is greater and vice versa. This is a matter of influencing standard deviation.
The ratio of nature : nurture is obscured and we don’t know exactly the influence of each. At what point is nurture overcoming nature? The question which follows is by who’s standard do we judge the virtue of indoctrination to create either differences beyond the norm or sameness beyond the norm? Doctrine variations by place and time are probably not whimsical but to suit a need or purpose of that time and place. Then of course we ask who’s purpose for what? Is it relevant in our place and time? The answer most likely tracks the worldview of the person providing the answer.