Completely off base.If I remember correct, he is the prince's adoptive brother but he calls her sister. Correct me if I'm wrong.
Claude was adopted into the Graupner main branch family as Remilia is the only child and is expected to marry into the royal family. Which means Claude is needed to become the heir of the main branch. I expect with that background, he would have been adopted in, even if he didn't become an orphan.
He is Remilia's cousin by birth and brother by adoption.