It makes sense from an evolutionary perspective. The vast majority of men across history died without fathering any children. Doing so was a very real competition to secure power and women and defend what you had from the marauders. Our ancient ancestors were the ones that won these competitions; the ones that lost either died in combat or died as celibate servants to the more powerful men. So even though monogamy and modern society in general has changed the rules somewhat, the instinct is still deeply ingrained in most of us.
Women of course competed with each other for the attention of the most powerful men, but when you think about it from an evolutionary perspective, ultimately all that mattered was that they and their children were cared for and protected. Even in the absolute worst case scenario, your tribe being conquered and all of your men slaughtered, you would most likely be taken as a wife or concubine by the conquerors and life would for the most part continue on as usual after some time. So it isn't surprising that women would be predisposed towards cooperation. Raising children means that you get far more from being able to get along with your tribe and integrate peacefully into a new one if need be, than "winning" any single small victory. And in turn, women that were obsessed with small victories were more likely to piss their tribe off and find themselves ostracized or exiled.
This explains the difference in male and female bullying behaviors as well. Boys publicly display dominance because it confers status. Girls do it far more subtly, in private, so that they can put other girls down without breaking the veneer of social harmony or changing the way others perceive them.