In English you would never say "his right" (except when naming a specific right), yet it is accepted. Would you ever translate "seinem Recht" to "his right" (singular) in the same circumstances?
I agree that I would not use "right" in singular here in English except the situation you mentioned. Duo probably accepts "his right" singular because the German sentence is singular - "seinem Recht" instead of "seinen Rechte". Sort of how sometimes we're given a German idiom that doesn't translate to English at all when you go word for word, but Duo accepts the word for word translation.
I think it's valid. The right must be implied by the context. Alone it would be odd.
Agreed. If the right of freedom of speech, for instance, were being discussed, you could certainly say, "He insists on his right (to freedom of speech)."
I'm not quite sure if you can say "seinem" in this sentence. I would say "Er besteht auf sein Recht"
The translation seems wrong. Er besteht auf seinen Rechten. He insists on his rights. Er besteht auf seinem Recht. He insists on his right.
For all of you who don't understand the "be" prefix..