I will try to answer your question as best I can based on my lay reading and my personal observations. I am an MD and care for more than a few trans patients and I work with several trans people too, both men and women and also have a non immediate family member who is trans. However, I am NO expert on the topic. However, you asked me my opinion, so I will give it to you. You will probably read this and scoff or say I am a slave to gender ideology, well, I am ok with that. I harbor no malice to women, an in fact I am known a good ally for women in my personal and professional life. I don't have seething anger or live in a basement lacking in love or human connections or whatever else you have accused some others of doing. I am not a troll. As I have said many times, only to face derision, much of what I know on this topic and what has informed my opinions has been taught to me by females, NOT males.
Ok, so I think gender identity is a core part of a person's sense of self that forms at a relatively young age. The sense of self is a real thing, it can be documented and measured and it not delusion or imagination. I am sure you have a strong sense of self, we all do. Anyway, this is a sense of self that is genuine, very deeply held, persistent and fairly immutable. This sense of self/gender identity deeply and profoundly informs how a person behaves, appears, interacts with other individuals and institutions and their general relationship to the outside world. It does not mean that they conform to some sort of stereotypical behavior associated with a given sex or gender. In turn this outward expression of the gender self also has a major impact on how individuals and institutions view and interact with the person. I am not sure of the determinants of this sense of gender identity, but I would not be surprised if there are biological ones, genetic, neural, developmental reasons. In the end, I also am not sure we can really ever know why people have a non conforming gender identity or if it is "real or not", but I am comfortable taking people's word for it and treat them accordingly. I am sure there is good science that could be done to explore all of this and I would oppose efforts to block a better understanding. The problem is that people have come to think that pursuing understanding is a form of assault, and this makes me sad, but that is not too surprising given the nature of a lot of the posts (I am not saying this is true of you.). There has been a complete loss of trust that is for sure.
As for what renders a male a woman or a female a man, it is having this sense of self and therefore navigating the world with this identity all day every day. A trans woman is most definitely not a man who simply cross dresses. A trans person will have many of the same experiences, opportunities, oppressions as a cis person of that gender, but not all of them of course. I do not believe a trans woman can have menstrual cramps or other biological phenomenon that are exclusively related to sex, but in many of the ways that count in society, I believe a trans man or trans woman have many of the same experiences as cis men and women of the same gender identity, and this is enough for them to be considered men or women for me.
I know you will hate my use of this quote, but I am going to use it anyway:
"One is not born, but becomes a woman."
I think the major social pressures/oppressions that led Simone de Beauvoir to utter these words apply to both cis and trans women alike. So I do think the quote is apt. I am not denying that females have unique biological features from males and vice versa, and I do not think that trans women are female or that trans men are male, but I do think they are women and men respectively as it pertains to most of our social structures, workplaces, etc.