The existence of intersex people illustrates that sex is biological and binary
Folk are intersex because of how their biology is within complex and varied manifestations of two sexes.
There is this rather tedious game that is sometimes played where the existence of intersex people is somehow taken to indicate that either sex is not biological or that sex is not binary.
Any suggestion along the former lines is easily dealt with: a person is intersex if they have a specific type of pattern of biological features. That is, in the words of the UN OHCHR (the UN Human Rights Office):
Intersex people are born with sex characteristics (including genitals, gonads and chromosome patterns) that do not fit typical binary notions of male or female bodies.
Those sex characteristics are, of course, biological. It is the biological structure of their body that makes someone intersex.
The wording typical binary notions of male or female bodies is a genuflection towards sex as a socially constructed category. That categorisation within a language is a social act does not make the thing being referred to thereby socially constructed. To act as if it does is to confuse use of a term with the referent of the term.
This use-and-referent confusion, this confusion between category and object, is not a case of failing to distinguish between use (“I like eating cheese”) and mention (“cheese has six letters”), but it is similar level of logical error. Both types of logical error come from the aboutness of language and thought; from us using language to speak, and categories to think, about other things.
Sex existed long before anyone was developing categories about sex. Sex continues in the biological world all around us, regardless of what categories we may choose to use, and how.
It’s all about the gametes
If your body is structured to produce small, self-moving (motile) gametes, you are male; regardless of whether any viable gametes are actually produced. If your body is structured to produce large, not self-moving (sessile) gametes, you are female. Also regardless of whether any viable gametes are actually produced. At its base, sex is defined by reproductive function. Such a pattern of only two types of gametes means  that sex is, at its base, binary.
While, in its base evolutionary function, sex is binary, the manifestations of the binary nature of sex in organisms can get quite complex. [That sex is binary doesn’t mean that bodies are.] In a way, that is probably the evolutionary point, as the most widely accepted hypothesis among biologists about why species adopt sexual reproduction via gametes is that it was an evolutionary adaptation to deal with pathogens. By sexually reproducing, the genetic die were being thrown again and again, giving sexually reproducing species a much better chance of having genetic lineages that could survive a particular pathogen.
In us Homo sapiens, as mammals, there is a set of characteristics that are specifically typical of the male-body structure and a set of characteristics that is specifically typical of the female-body structure. If you have some characteristics from both sets, you are intersex. But it is precisely the existence of these two sets of sex-typical biological characteristics that creates (1) the possibility of being intersex and (2) enables identification as intersex.
So, the existence of intersex people does not confound either the biological or the binary nature of sex. On the contrary, it refers to a set of people with various patterns of biological characteristics that can only be identified as falling within the set of intersex people because of the binary and biological nature of sex.
Arguments about, for instance, the concept of binary being binary — something is either perfectly binary or it is not binary — are ways of avoiding grappling with the biology. Defining binary in a way that means nothing biological of any complexity can meet it is not a proof that sex is not binary. The small self-moving gamete/large not-self-moving gamete difference is binary in the sense that counts in terms of evolutionary function: reproductive function. Reproductive function that is subject to, and shaped by, evolutionary pressures.
It is that evolutionary pressure that makes, sex, in its base evolutionary function, binary. Given certain basic conditions, if a species reproduces through the combining of gametes (i.e. reproduces sexually) then having two types of gametes — small, self-moving (motile) gametes and large, not-self-moving (sessile) gametes — is the only evolutionary stable outcome. Hence, in our biosphere, sex is binary because there are only two types of gametes.
Thus, there is no third sex [at the level of gametes]. There are neuter forms of females in eusocial insects. In some species, an individual can change sex. But there are only two sexes [in the sense of only being structured to produce one of two types of gametes]. Some individuals partake of characteristics typical of both sexes. That does not make them a third sex.
[If there were actual hermaphrodites in a species with males and females, there would be grounds for calling them a third sex, as their bodies would be structured to produce both gametes. That would not, however, change the binary nature of sex in its evolutionary function.]
The key confusion is failing to grasp that the binary nature of sex applies to its evolutionary function. If conjoining gametes is how reproduction happens, and there are only two sorts of gametes in play, then sex is binary. It is that simple.
This is NOT a claim that individual organisms cannot have a mixture of features. It is not even a claim that individual organisms cannot move across the boundary from one sex to another. It is also not a claim about animals conforming absolutely to to two, and only two, physical structures. It does not even preclude an organism producing both types of gametes, either sequentially or simultaneously.
The binary nature of sex is not defined from structures of bodies inwards. It arises from reproductive function outwards. As a biological process, reproduction has consequences for physical structures, but these can be quite complex and varied. A complexity and variance that does not in anyway change the binary nature of sex, though it does considerably complicate its expression in biological structures.
In summary, there are only two sexes [at the level of basic reproductive dynamics], defined by there being only two types of gametes. There is no third sex [at the level of reproductive dynamics] because there is no third type of gamete. Hence, sex is binary, however complex the manifestations of that underlying only-two-types-of-gametes pattern may be.
[So, when folk say that sex is binary, what they should mean is that there are two types of gametes. And when folk say that sex is not binary, what they should mean is that the biological expression in actual bodies of the binary nature of sex is not itself binary.]
The rest is just tedious word games, with more than a dash of logical confusion.
[The bits in square brackets represent adjustment in the light of some helpful criticism in the comments.]