I honestly find this trend towards nothing meaning anything so fascinating, if also a little bemusing.
I'm a black man, which means that my skin is dark and I'm of Afro-Caribbean parentage. Is it false, or science denying or in any way controversial to say that I'm black and somebody of Scandinavian heritage for example, isn't?
I mean, I agree with the sentiment of the point you make at the beginning; what does it mean to be black? Being black doesn't say anything about my character, or my intelligence, or my ability to dance. But it does say something about my ancestry, and my genes, and the chances of me having sickle-cell trait.
The very fact that there isn't a single defining experience of being black is exactly why it's nonsense to say that you are black if you feel like you are. Being black doesn't feel like anything. If you "feel" like you're black, you're projecting an imagined view of what being black is onto your reality.
What next? Trans-height? With people below five feet fighting to end discrimination by the NBA? Or trans-weight? With people who weigh 500 pounds fighting to end the unfair way they're singled out by gravity?
We can let things be what they are, and still honour people's right to live their lives as they want to live them. If somebody wants to put their hair in dreads, or wear a dashiki, or eat okra and fried chicken, more power to them I say. I think the concept of cultural appropriation is equally questionable. But if they think that doing these things makes them a different race, they are, to put it kindly, confused.