Doesn't your last bit about birds rather demonstrate the point that to be a bird one has to have the biology of a bird. It the biology that makes the bird, not people performing aspects associated with birdness? Or even an inner sense the bird has that it is a bird. If it had scales and four legs, would it be a bird?
Similarly one can choose to adopt characteristics society may associate with ' 'womaness' but that does not make one a woman. The only thing that makes one a woman is female biology.
Gender identity is something that was invented to get around the inconvenient fact that woman means, and has meant in probably all languages, in all human history, adult human female. This means that people who are male, did not fit into this category. Therefore, to allow them entry to 'womaness' the definition of women needed to be changed. And hence the unevidenced, unscientific, undefined notion of ' gender identity' was born. Gender identity was not discovered, it was created to solve an ideological incoherence.
Gender identity doesn't mean anything, of course. And, importantly, it doesn't explain why women have been so disadvantaged and discriminated against throughout history, in the way that the definition of woman based in our female bodies does. That is because of our reproductive function, the fact that straight men want to have sex with women, the fact our bodies are smaller and weaker than men's so men are able to abuse and attack us. So removing the reality of our sexed bodies from the word woman, no longer gives us the means to understand the foundation of the oppression and harm that has been done and is being done to us- and without a proper fact based analysis of our oppression, and without a word to describe ourselves that reflects the bodies that our oppression is rooted in, we are handicapped in being able to name, describe and fight our oppression.
But don't let that bother you.