The wonderful thing about not reflexively constructing things but investigating them as they are is that questions like this have definitive answers.
The spotted hyena, Crocura crocura, is a member of the suborder Feliformia--the cat-like carnivores--of the order Carnivora. Both Feliformia and its sister suborder Caniformia (the dog-like carnivores) have inherited the mammalian sex-determination pathway and tissue sensitivity to sex-specific hormones.
The birthing animals are XX, create large nonmotile zygotes, are impregnated by XY animals, and have mammary tissue from which they produce milk and nourish their young. From evolutionary homology, evolutionary genetics, and extant traits, these animals possess most of the usual sex-specific traits characteristic of female mammals. They do have very high levels of androstenedione, which seems to be responsible for some traits that are more frequently male-specific sexual dimorphisms appearing in the females instead/as well. The other ones--the hyenas who make sperm--possess almost all of the usual sex-specific traits characteristic of male mammals.
Unless you're Freud, or not paying much attention, the phallus-organ of the female spotted hyena is not going to confuse you. At all. Female spotted hyenas are unambiguously female; male spotted hyenas are unambiguously male.
Yes, female spotted hyenas have unusual genital morphology to the point where it's not entirely obvious what to call the female's structure ("pseudo-penis" was chosen--fair enough, though not the only possible choice). But it is what it is. I don't think the female hyenas lose a lot of sleep worrying about which words we choose. They lose sleep worrying about the endless conflict with lions.
Because biology is a weird and wonderful collection of accidents that just happen to work (except when they occasionally don't), we always have to look for overall similarity, not get fixated on one or two things, when categorizing things. There is an overall similarity there, in many many places. Hind limbs of humans and seals and horses and wombats are tremendously different in detail but there are also tremendous similarities that tell us that, in fact, these things are all alike in some very important ways.
If you have species that produce both motile and non-motile gametes in the same animals, like slugs, it doesn't make sex constructed. It makes slugs different from mammals, though! If Saccharomyces cerevisiae has a and α mating types, it doesn't make sex constructed. It makes yeast different from mammals, though! If male birds are ZZ and female ZW; if crocodiles don't even have sex chromosomes but the sex determination pathway is temperature-dependent instead; it doesn't make sex constructed. It means birds, crocodilians, and mammals each have their own regulatory system to control an ancient dichotomy.
There's no point referencing old ideas about the nature of female vs male zygotes or individuals or whatever, save to point out that we humans love to tell ourselves stories about things instead of investigating the heck out of them and just describing what's there. There you go again, humans! Got your presumptions confused with reality!
Once we come to our senses we are called simply to drop the foolishness. We should not pretend that because we were once foolish, there is nothing but foolery to be had. Sometimes underneath the construction there is literally nothing, nothing more profound than whether hair ties or scrunchies are the cooler way to hold a ponytail. But usually--at least when it comes to widely-conserved biological processes--there is a lot of actual structure there that we'd better come to grips with if we're going to use knowledge to make better lives for ourselves and each other rather than use dogma to control and manipulate each other.
The non-constructedness of secondary sexual characteristics (including cognitive ones) is one of the most important realities for trans people, because there is nothing else that can answer the right-wing claim that it's all just "transgenderism". Tiny numbers of impassioned people do not have a free pass to restructure society. Society is for the good of as many as possible, not for the good of the few who throw an ungodly temper tantrum demanding that everything be reshaped to fit their vision. According to the right, that's all there is: mental illness and toxic ideology; all construct, no reality, nothing there.
But what if it's not all just arbitrary human constructs? What if most people develop either as men or women, view themselves as men or women, respectively, and are attracted to women or men, respectively, but a significant number don't? In an affluent society we have the space to ask, without incurring massive negative consequences for even considering it: can we find ways to admit that there is that variation and enable even more people to live satisfying, productive lives by taking it into account rather than pretending it doesn't exist or stomping on it as viciously as possible whenever there's a hint of it?
The gay rights movement argued extremely effectively that homosexual desire is at least in major part not constructed. Rather, it is biological. This has been absolutely central to the argument that sexual orientation is a matter of human rights. There are massive amounts of social construction around sexual desire of all sorts, but the rights obtain after you dig through all that and recognize that, actually, it's not all constructed. You can't pray the gay away when the gay wasn't responsive to human social pressure to begin with (and divine intervention is not routinely forthcoming). The argument wasn't successful primarily because it sounded good. It was successful primarily because it seemed to comport well with reality.
Sex is ancient. It's heavily conserved. In humans, it's strongly (but not completely) bimodal. None of that is arbitrary; all we can "construct" is a petulant unwillingness to look at the patterns that are there (either denying the exceptions or denying the rule). We can't even construct the physiological consequences of XXY; it is what it is, and the mostly-like-XY-male-but-with-some-characteristic-differences is what it is regardless of how we talk about it.
Gender, as defined as the social expectations regarding people of different sex, is by design a constructed thing. It might be reified to some extent by sexually dimorphic cognitive machinery, but even if it is, there are still vast swathes that are simply convention that might work as they are and also might work fine otherwise. Untangling which is which is a major, important undertaking.
Confusing (a) the fact that we can layer assumptions, presumptions, demands, terminology, and so on, on top of sex too, with (b) there being no fundamental difference in the constructedness of sex and gender, is just that: confusion. Not only is it a confusion, it's a pernicious confusion for trans rights because society constructs things as a whole, not just as individuals, and a lot of people sure do wish that trans people just got constructed away into "men are men, women are women, and everything's all simple again".
It is in the lack of constructedness of sex and self-identity of sex in which all the best hopes lie for expansive rights.