Your question reads as disingenuous. In the case that you aren't: Sex can be determined biologically. Gender has a significant social component, and can't be determined through biology. For example: "Why is lipstick in the women's department at the store?" is not a question that can be answered through biology.

However, if you want examples of non-binary sex, that exists too: it's called intersex.

Intersex people are male or female. Male/female comes down to gametes and intersex does not somehow prove that biological sex is a spectrum.

"rare" doesn't mean "never". Caster Semenya is an Olympic candidate who is getting disqualified from some women's racing events because she is intersex and has just a little bit too much biological masculinity to compete as a woman.

See https://en.wikipedia.org/wiki/Intersex

I don't think anyone's claiming that "biological sex is a spectrum", but some people "do not fit the typical definitions for male or female bodies". Some are born with ambiguous external genitalia, but there are also people who look completely female while having male (XY) chromosomes. Since various distinct medical conditions are involved, it doesn't make sense to call it a "spectrum" even from a medical point of view.

Spectrum doesn't mean 1 dimensional.

Some people don't produce gametes.

The scope of the biological study of sex is, in fact, broader than the presence of a Y chromosome.

“Males are males” is what you said. I think you may have been conflating gender and sex. The reason why “gender” and “sex” are different words is so we can discuss them separately—gender is social, sex is biological. If you’re an anthropologist, you might study gender norms in different cultures. If you’re an ichthyologist studying fish, you might study the sexual characteristics of different species.

Ichthyology is interesting. Clownfish, for example, are born (sexually) male. In a given group of clownfish, the largest will become female. This is a physiological change. However, it’s not “genderfluidity” because clownfish, like most animals, do not have gender.

Gender, since it is a purely social construct, is somewhat arbitrary to begin with and depends on cultural norms. Saying that sex and gender is equivalent is asserting that a particular set of cultural norms is correct and other norms are incorrect, and it’s a common anti-trans talking point. If you got downvoted, it’s probably because you were repeating some of those tired talking points—in your words, “males are males” and “the differences in the brain could be described as a mental illness”.

Gender dysphoria could be considered a mental illness, but it could equally be considered a social / environmental problem. From a harm reduction standpoint, the general consensus is that it can be easier for someone to transition to a different gender rather than try to treat gender dysphoria itself as a pathology. In other words, is gender dysphoria pathological, or is it a manifestation of a problem in the environment? Based on the evidence from treatment outcomes, it seems obvious that treating gender dysphoria as a mental illness is not clinically useful, at least in general.