It never ceases to baffle me that there still are people who insist that there are only two genders because sex is binary because there are only two types of gamete (in humans, there are life forms with more than just two gametes). I am especially disappointed in Dawkins, a writer whose popular scientific books I really enjoyed and who, as a biologist, really should know better.
Trying to conflate “sex” as it refers to whole persons with “sex” as it refers to gametes is a prime example of a bad kind of scientific reductionism. Just because there is one word – in this case “sex” – does not mean that it means the same thing all the time, everywhere.
Firstly, sex at birth is not assigned according to any kind of gamete that an individual produces, it is assigned as a best guess based on external genitalia at birth. As such, it is mostly right, but there are cases where it cannot be ascribed with confidence and also cases, where it later shows being wrong. And cases where surgery is actually used to shoehorn a person into one of the two boxes.
Secondly, there are a lot of people who never produce any kind of gamete their whole life. What sex do these people belong to? If one decides that sex must be a binary based on the type of gamete produced by an individual, one must then decide that these people do not have any sex whatsoever. This is the exact point where the concept of binary sex when referring to people and not gametes breaks down. There are a lot of further complexities, but this suffices to disprove the idiotic notion that people can be sorted into exactly two categories based on gametes.
And that’s without going into the whole concept of gender, which has nothing to do with just biology. Gender is a linguistic/social construct. Just as gametes are just one criterion in determining a person’s sex, a person’s sex is just one criterion in determining their gender.
So although sex as it pertains to gametes is binary, sex regarding whole people is a bit more complicated. The word sex cannot mean the same thing in both cases and does not need to have the same constraints. Sex, when referring to whole individuals, is not binary but bimodal. Which is similar, but not the same.
Wishing for reality to be simple because one specific language (in this case English) has just two words for gender and wishing to shoehorn everyone into those two words is akin to insisting that the rainbow has a limited number of exactly distinct colors because we have assigned distinct words to some bands of wavelengths.