I don't believe your identity as a woman is compromised by another woman's identity.
My identity formation was based on the known stable things of having a brother.
en.wikipedia.org/wiki/Identity_formation
Identity formation, also known as individuation, is the development of the distinct personality of an individual regarded as a persisting entity (known as personal continuity) in a particular stage of life in which individual characteristics are possessed and by which a person is recognized or known (such as the establishment of a reputation). This process defines individuals to others and themselves. Pieces of the person's actual identity include a sense of continuity, a sense of uniqueness from others, and a sense of affiliation. Identity formation leads to a number of issues of personal identity and an identity where the individual has some sort of comprehension of themselves as a discrete and separate entity.
Identities are formed on many levels, micro, meso, macro, and global. The micro level is self definition and relationship to people and issues as seen from a person or individual perspective. The meso level is where our identifies are viewed, formed, and questioned from our immediate communities and/or our families. Macro are the connections among and between individuals, issues, and groups as a view from a national perspective. Lastly, the global level is connections among and between individuals, issues, and groups from a worldwide perspective.
So when my brother decided, without warning, that he was now my sister this had a rather profound affect on my own identity.
I can not help this, because of how identity formation occurs.
It affects not only your own identity, but also how you relate to others
'Do you have any brothers or sisters?'
Its an innocent and bland enough question for most people. For me its now a sharp intake of breathe and trying to decide whether i go with 'its complicated' or decide whether to explain. The reason people ask you the question is because they want to relate to you 'oh i have a brother too' or 'oh thats how girls and boys get on as children'. So your lived experienced is a pretty big deal in this context.
Then there's your place in the family. Being the eldest. Being the only daughter. Whether you are equal or come second to your brother's identity and told to submit to their belief of what a woman is.
Then there's how you relate to other women in terms of all those other lived experiences as a woman. Those shared experiences in your formative years are important.
Its funny, because when you explore this subject, it soon become apparent that this ideology is utterly toxic because of how many families it destroys. Its like a grenade being thrown into a family.
Family identity are very much intertwinned and can not just be changed without having an effect on other members of that family. When ones identity is more important than anothers and others are forced to deny or erase their past in language and how they relate to others, this is something of a big deal.
This isn't bigotry, this is a natural response to a distruption in identity formation in others.