Your argument builds upon two assumptions: 1. There is a finite number of pre-defined gender-identities which are not interchangeable. 2. Identity is bound to biological organs

I see no proof of these assumptions. How humans see each other and themselves is not primarily chosen by their unchangeable (save medical interventions) organs but by how we are read by our peers. We are not talking about a purely biological problem, but a social one. Humanity has, for the longest time being, assigned fixed profiles to these readings, which luckily has started to break up in the last decades. Now of course each individual is free to base parts of their own identity on their unchangeable biological traits, and you also are free to do so, but that does no generalize to the whole population. Humans should be free to base their identity to their own liking.

This can lead to some cases which do not make sense like your "male menstrual cycle" example, granted. These anecdotes however as well do not generalize towards the initial claim of the other user and the first assumption.