And the Duolingo version (wychowujesz twoje dzieci) sounds very awkwardly… I believe it is a mistake, and should be reported. Wychowujesz swoje dzieci is the proper Polish way to say that.
swoje puts emphsis on 'person's own' - Ja mam swoje zabawki, ty masz swoje == I have my toys, you have yours, (and I don't need yours or I don't want to share).
No, you just gave a sentence that puts such emphasis, as your example. And in English it would, I believe, work exactly the same way: You have toys vs you have your toys.
Polish just uses swój as one’s own in such sentences, eg. dbaj o swojego ojca would just mean: care for your father, and I don’t feel any emphasis on your here (although, one could in fact just say: dbaj o ojca, which I’d translate as care for the father). But saying dbaj o twojego ojca would, in my opinion, be awkward.
my logic went along the Croatian logic, because we have the same pronoun and it definitely does not work as an emphasis (apart from sentences similar to what you gave as an example). but then again, logic of one language does not have to be consistent with the logic of another.