My opinion is that Duolingo strings the words together itself, and so to make it sound more fluent it cuts out a little at the start and end, probably to eliminate blank space, but ending up taking out some of the word sounds. This is very evident in sentences with a lot of small words (et, en, og) as it sounds very broken and unnatural.
From what I understand, 'barnet sitt' would imply that the "him" we're talking about 'owns' or has the child. But, since it is 'barnet hans', we're talking about "that guy over there" or "some other guy's" child.
Edit: Mhm, now that I'm thinking about this even more...and I could still be wrong... The subject of the sentence is "the child". 'Sitt' can only be used to describe something the subject has or owns. Obviously, we're not talking about the child owning his/herself. So, we need to use 'hans' to talk about "his child".
But, let's say, we wanted to say this sentence: "The child does not like his (own) dogs." Then we would say/write: "Barnet liker ikke hunder sine."
"Sitt" is a reflexive possessive, so it refers back to some previous subject. Note that here, "barnet hans" is the subject. There is no "previous" subject. In contrast, I might say "Han liker barnet sitt". He likes his child. The subject here is "han". So "sitt" refers back to this earlier subject. If I said "Han liker barnet hans", it would imply that he likes some other man's child.
Barna hans liker ikke ulver would be correct, though, if it was "his children," right? Is there any way to hear the difference, or do they sound exactly the same, and you just determine (IRL) what someone is saying based on context? Like, you might know if a man has one or more children?
Yes, "Barna hans liker ikke ulver" is a valid sentence. "barna" and "barnet" should not sound the same, and so you should not have to rely on context. "barna" should sound like barn-ahh and "barnet" should sound like barn-ehh. With really well-enunciated audio I can hear the difference, but in this particular exercise it still sounds more like "barna" to me.