Here's what Dawkins says about sex in The Selfish Gene:
There is one fundamental feature of the sexes which can be used to label males as males, and females as females, throughout animals and plants This is that the sex cells or 'gametes' of males are much smaller and more numerous than the gametes of females. This is true whether we are dealing with animals or plants.Male whales are the ones with the numerous small gametes and female whales are the ones with fewer, bigger gametes. Same goes for all animals and all plants, if Dawkins is to be believed. And here's what Dawkins has to say about race, in a chapter of his book The Ancestor's Tale. He says a race is like a sub-species, where the differences separating different subspecies are superficial and brought about when sub-populations live in different environments. (He makes some of the same points in this article.)
Of course there's a whole lot else that we think about sex and race. Historically and still today, people represent sex and race as involving very big differences; and as being "essential" in the sense of profoundly influencing an individual's life; and as coming with associated norms, so that men are supposed to act one way, women another, or one race is supposed to lord it over the others. A lot of those representations and norms are just plain false or ethically misguided, if sex and race are as Dawkins says.
But take note! Being critical in that fashion is of course an option for a biological realist. So there's nothing particularly anti-progressive about realism. In fact, I would go further. If I want to liberate myself and all women (for example) from the shackles of old norms and stereotypes, the language of falsehood works really well. Women aren't like that. Those norms are groundless. If social constructionism opens the door to saying certain ideas and norms aren't "inevitable" so does good old fashioned criticism of theories for being false, or norms for being inappropriate.
So what makes people prefer to speak of the social construction of sex, if they're critical of traditional stereotypes and norms, rather than speaking of the misrepresentation of sex? Why do they think sex must go away entirely, no longer being seen as part of the pre-existing, mind-independent "furniture of the world"? Likewise, why do they want to talk about race being constructed by the racist, instead of saying it's horribly misrepresented?
An awful lot of the argument for social construction seems to have to do with grey areas, intermediates, borderline cases, vagueness, etc. The biologist Anne Fausto-Sterling claims that about 1.7% of babies are born intersex. In each society those babies get sex-labelled in ways that have to do with the representations and norms we have regarding sex. Fausto-Sterling makes a convincing case for that. Why, though, should we conclude that the sexes of the other 98.3% are socially constructed?
The same thing seems to go on in the debate about race, except that the grey area is much more extensive due to intermarriage. People who are in fact mixed-race get race-labelled (by others or by themselves) in ways that have to do with the full set of representations and norms we have regarding race. Does that mean that all of race is socially constructed? Why would anyone think so?
In fact, vast numbers of our concepts (maybe even most) involve grey areas in the same way that race and sex do. There are clear cases of tables and clear cases of chairs. Then there are stools. A big enough stool is really a table. But what about medium-sized stools? Are they tables or chairs? There might be customs and assumptions that people use to settle such question, but so what? The reality of two categories can't be wiped out just because there are numerous entities that lie in between, and just because all sorts of norms and representations help us decide where to put the inbetweeners.
There is another reason why some people want to do away with biological sex as a reality "out there." Grey areas aren't a problem for frogs, kangaroos, whales, and apple trees. But people have strong feelings about the way they're sex-labelled. If an intersex person lives happily as a woman, it may be offensive to her for anyone to say she is in fact intersex, and not female. Likewise, even in cases where labels apply clearly, there can be preferences at odds with the labels. A transgender woman may prefer to think she was not born biologically male.
I wonder, though, whether any theorizing about sex and race that tries to be sensitive in this way really deserves to be called philosophy or metaphysics. How could the truth about what sex is turn on how people feel about how their sex is represented? And if we do take into account such sensitivities, where will we wind up? It's certainly going to be a pretty weird philosophy or metaphysics that sees sex in humans one way, but sex in the rest of nature in another way. Or that sees sex in all of nature in a way that's ultimately motivated by sensitivity to certain humans.
Of course saying sex is real, and not socially constructed, doesn't settle every question we might care about. People don't just have sexes, they also have sex-identities. (I agree with Heath Fogg Davis—in the book Beyond Trans— that this is a better term than "gender identity" for many purposes.) When someone has a male sex but a female sex-identity, what then? To my mind, this is an ethical question (and the answer is that sex-identity takes precedence). It's a question about well-being and how to treat people. I don't see how it can makes sense to let progressivism color what we think about what categories like race and sex are in the most fundamental, metaphysical sense.