I’m aware and I agree—that’s (usually) how biology works. However, it seems like the author’s reasoning for changing the dragon lore has more to do with animals and their perception of gender/gender binary—or lack thereof. (The author isn’t denying the existence of biological sexes—although, tbh if they wanted to they could, because it’s their *fictional* story.)
Animals don’t have a concept of gender or literally anything other than how the author describes it: “there are some dragons who lay eggs, and some dragons who fertilize them.” Animals don’t differentiate between the two sexes beyond how it’s said in the quote.
I’m sure there’s more that could be discussed regarding the perceived level of the dragons’ intelligence, but those are unnecessarily complicated anthropological concepts, and it’s also ultimately up to whoever is creating the lore.
Reproduction is a foundational element of life. It is inconceivable that any sexed species, nevermind an intelligent sexed species, could evolve without developing an awareness and concept of a biological phenomenon that is integral to sexual reproduction: that individuals come in two types with differing roles, one of each being necessary. This need not imply anything else: things like grammatical gender and gendered behavioral norms need not necessarily follow from sexual differentiation. But they would surely be aware of such differentiation and have a vocabulary to describe it.
It feels like the author decided they wanted a non-binary species for political reasons without considering what that actually means. They could have created a species where biological sex simply does not matter beyond the rudiments of sexual reproduction. Members of such a species would not self-identify as non-binary: they would self-identify as male or female and barely give the matter further thought. Self-identifying as non-binary is a conscious denial that the binary applies to you, which would be insane if referencing biology; it only makes sense as a rejection of something else, namely, norms associated with a person's sex. An entire species would self-identify as non-binary only if they first developed gendered norms, then chose as an entire species to deny those norms, and were still living in the shadow of that denial. I should like an explanation of how such an event came to pass, although I doubt anyone could create one that I'd find plausible.
Of course, I doubt that was really the author's intent; they probably just wanted a species without gender norms. It is ironic that they chose to do so by making it "non-binary," as that very concept implies said norms and therefore perpetuates the species' subjugation to what the author was trying to eliminate.
I never said that they're unaware of there being differences between the two sexes, just that their awareness of it still doesn't include the concept of biological sex beyond "dragon type A has this, dragon type B has that; when the two types mate it produces offspring".
If this was a real life situation, the level of self-awareness and intelligence that the dragons have would probably include an understanding of biological sexes, and frankly, probably some understanding of gender/gender identity as well. As you said, "...they would surely be aware of such differentiation and have a vocabulary to describe it"---sure, I agree, if this hypothetical situation was following the "rules" of life IRL. However, that isn't the case and the IRL rules don't actually have to apply in any capacity, because it's a fictional story in a fictional world.
Clearly it's possible for the details of the fauna/flora of an imaginative world to become blurry. The author never said the dragons self-identify as non-binary, so much as state that they just don't identify, period. They specifically said that the dragons don't classify themselves on a binary---as in, for them there is no binary, not that there's a binary that exists that they don't include themselves in. I would go so far as to suggest that maybe the author specifically didn't refer to the dragons as being non-binary, because---as you said---to do so would imply the existence of a binary within the species in the first place. It seems like you saw "no binary" and assumed the author meant "non-binary" and created the logical dissonance that you're disagreeing with in the first place.
Though, again, there's no requirement for the story to follow logical boundaries because it's entirely fictitious.