"His sister is vegetarian."
Translation:Hans søster er vegetar.
"sin" is kind of like a reflexive possessive pronoun, which is used for singular, third person things like "Han spiser sit æble". If you wrote "Han spiser hans æble", it would sound like he's eating another man or boy's apple and would need to be used in context. Here it can only ever be "hans" because it is part of the subject (the first noun in Danish)