I believe you are asking: If gender identity and racial identity are both social constructs, what makes them different types of social constructs, and, specifically, why should we acknowledge that one can assert one's own gender identity while insisting that one must accept one's given racial identity?
Regarding the part of the question that focuses only on racial identity, there is an existing discourse. Very broadly, it has to do with cultural appropriation, with history, and with today's realities of what it means to be Black in society. I am white, and I am not the best person to explain these topics. I provided 8 links in my article to other people who have explained it. Generally, I find their arguments persuasive, or else I feel I have no grounds on which to dissent. Generally, I will defer to what seems to be the consensus of Black people on the question of white people presenting themselves as Black. It would be racist of me to challenge them on that.
If you want more information on the discourse over the small number of white people who claim to be Black, you might start with the article links I provided. From there, you will be equipped to find more.
My concern is simply that the argument over Rachel Dolezal's life choices and professed identity not be used, please, to delegitimize transgender people. Not only is Rachel Dolezal just one person, she's not even transgender. Her experience just IS very different from transgender people's experience. I want to leave it at that. Especially because the "transgender-and-transracial?" discourse is used to delegitimize transgender people far more often than it is used to affirm them. And also because many Black people have explained the problem with the choice of a white person to present themselves as Black. The comparison is bad for everyone, and I don't want to dig further into the details.
The point of my article was just to demonstrate that Richard Dawkins was, in fact, tapping into an existing discourse that goes back at least to 2015. Some people had claimed he was innocent and could not possibly have known what he was referring to. I maintain that he knew, or could have and should have known, because the discourse does exist. He is responsible for the actual meaning of things that he says.
If we want to know what a transgender person thinks and feels, we could ask a transgender person. Coupling the topic with Rachel Dolezal is counterproductive. That is my stance here.