"It only concerns him and his family."
Translation:Det angår bara honom och hans familj.
Well, this is a good question. The answer is no, because "han" is not the subject of the sentence.
I'm not sure, but I think "sin" would have to refer back to the subject of the sentence, which in this case is "det" (i.e. not ham <- honom).
Technically, it depends on the context; but without context, I think that any reasonable person would interpret in the sense that warrants a sin.
Berör borde vara accepterat då det är något i familjen och kan vara personligt nära.
There is another good word other then angår in Swedish used for concerns in this context: Berör, this should also be right. Berör = concern, touch, feeling.