She loves the "family" godfather music plays in the background
I capitalised Family in my answer!
Didn't try, but "...her family." would work, too, right?
Then the Danish sentence would have to be "Hun elsker sin familie"
As far as I know, in Nordic languages, "familien" structure is used more than "sin familie" to mean "his/her family".
"familien" could mean any family. If you mean her family, "sin familie" would be way more common
Or also "Hun elsker hendes familie", if it was somebody else's family (of a female person).