I was shocked when I found out that gender identity is supposed to be an abstract feeling not at all based on the actual sex of one's body. If it happens to correlate with the latter, then you are cis and very privileged (more than, say, Caitlyn Jenner), and if it does not, then you are trans or nonbinary and very marginalized and oppressed.
I am quite sure my definition of my gender (identity, if you wish) is very firmly based on the fact that I have a female body and that others treat me a certain way because of that body. But though every person is supposed to be allowed to decide on their own gender identities, this does not extend to me or anyone else who defines their gender on the basis of their sex.
When it is 'people' who menstruate, then clearly my gender identity, based on the female body, is invalidated and I am excluded and so on, but because I am seen as 'cis,' and privileged, none of that matters.
Many seem to think that 'cis' just means 'not-trans' and that it is an ok label to accept, but that is an incorrect interpretation. The cis concept forces you to accept a gender identity not based on sex, and if you don't spot that, you will end up skating on weak ice when it comes to later inclusiveness demands.
I also like Saxby's take.