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.