Translation:It only concerns him and his family.
Would we not use "sin" because "honom" is the object and not the subject?
Yes, in order for us to use sin, it must refer back to the subject of the sentence, so since the subject here isn't han but just det, we must have hans instead.
Han pratar bara om sig själv och sin familj – He only speaks about himself and his family
Here we use sig because han is the subject.
"It is only about him and his family" was marked wrong
Is this wrong because angår as "concerns" is specific to the English "concerns" = worries them vs "concerns" = pertains to them? If angår is ok as "pertains to" is it also ok as "worries"?