The one thing women share is belonging to the sex (or being on the developmental path towards that sex) which typically, but not always, produces large gametes.
The female sex is the sex with a reproductive system organised around the ability to give birth. It doesn't matter, in terms of how women are treated by their cultures, that not all women will ever menstruate, say, or give birth. To others being female means that these possibilities will be taken into account in how societies treat women and girls.
Sex-based discrimination and sex-based oppression are also something which all women share, at least in the sense of shared history of what it has meant to be a woman (usually exclusion from full rights of inheritance, fewer rights in marriage, exclusion from many professions and higher education).
This sexism is relevant not only because this is women's shared history, but also because even in the most egalitarian societies and even for those women whose social ranking otherwise is high (in that they belong to the dominant, race, ethnicity, class, religion etc.) what I call 'the ratchet effect' works:
Pick a man and a woman otherwise completely identical, and ask yourself which of the two would be most likely to be heard in a debate, which of the two would be most likely to be hired or promoted into important positions, which of the two would be most likely to do the unpaid work in their families, which of the two would face higher risks of being sexually assaulted and harassed, and your answer might tell you what all women probably share.
But there is much that women do not share, and many women suffer from multiple forms of oppression and from exploitation not only on the basis of sex but also on the basis of race and/or class. Women also share allegiances with men of their own cultures and within their race, class and ethnicity, and these allegiances may contradict their allegiances to other women. Some women are far more privileged than other women, just as some men are far more privileged than other men, and all this complicates seeing the answer to the question in this thread clearly:
Women have shared interests in politics, even though we also have many interests which are not shared. For the first group, we need to organise as women (while taking into account intersecting types of oppression and their impact on the forms sexism and misogyny take), and for that we need to keep our name for our group. It's women and girls.