... then why isn't it called mano-kaya or kaya-mano?
Should there be an 'and' in there, or is an 'and' implied? Or could it be that one term is actually being defined in relation to the other?
This existence has two components, there are names and there are forms, a person is a mixture of both. What we call I
is a bundle of concepts and matter.
I believe rupa is more accurate than kaya because some Deva have a very subtle rupa, that might not fit into the notion of Kaya, and so rupa was more generic and inclusive. And nama seems to me more accurate than mano, for the same reason. Either of them are just conventional words to describe our existence in common language, probably more to those with less understandig of the Dhamma.
While accepting that there is no self in that nama-rupa, the person's existence is still real, so what can it be called in common language, in a way that does not mislead those who listen? There is no self, but there is an existence that was called nama-rupa by the Buddha. If he decided to call it that, and not any of the others, I sense he had a good reason to do so, avoiding confusion with other concepts.
In my understanding the extreme cases of the gods without rupa, or the case of the gods only with rupa, is left out for simplification of speech.