There was a time (not terribly long ago) when it was the conventional wisdom that the word “man” meant “male person”, and that both of those expressions specifically meant XY chromosomes, external genitalia (penis & testicles), no periods, no uterus, grows copious body and facial hair at puberty, etc.
Likewise in that same vein for “woman” meaning “female person” and corresponding to an array of physical genotype and phenotype characteristics.
We were introduced to the notion of transgender people via the personal narrative of people who explained that they should have been born as the other sex, that who they actually were was the same personae, the same identity, as the identities shared in common by the other sex. And they then sought physical transition in order to bring their bodies into alignment with their identities.
Society reacted to this initially with an attitude akin to “ooh circus freaks! weird kinky perverts!”, then a long period of somewhat accepting (to varying degrees) the people who identified in this fashion but still regarding them as irrelevant data as far as how they continued to understand what it means to be a man or a woman; and, now, in more recent times, by being increasingly inclined to fold an understanding of trans people into our everyday understanding of sex and gender.
At no point in this narrative so far does “REAL” crop up. There isn’t a “REAL” (empirically, objectively correct) answer available to us, precisely because the subject matter is social. The meaning of terms isn’t a fact that biological science, neurological science, embryological science can answer.
And we have not “arrived” at a final social destination whereby we can look backwards with smug self-righteous assurance that our current understandings are correct and that we were biased and wrong in the past. It’s still a work in progress.
“Hate speech” is also a social category. Nothing is intrinsically hate speech aside from how we collectively choose to define it. I, personally, would be fine with a board policy that it is not okay to use the loaded language “is not a real woman” / “is not a real man” but to allow and encourage discussion of the best use of various terms such as “man”, “woman”, “male”, “female”, the appropriate or inappropriate way to think of and understand transgender people, and where one’s feelings and reactions come from and so on and so forth.