Shouldn't it be sin 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.
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.
I don't think berör and angår are synonyms. I think berör is literally touching, and that angår is for matters that are important to you, that concern you (government rules for example). Perhaps a native Swedish speaker can confirm, please.
How do you choose between sin and hans
"Sin" can only be used to refer to the subject of the sentence. This subject is "det", not "han" or "hon."
Berör borde vara accepterat då det är något i familjen och kan vara personligt nära.
A couple of questions before, it's been said that "bara" has a special place and can be used before the verb. Can it be used in here like "Det bara angår honom och hans familj"?