"My grandmother likes milk."
Translation:Моя бабушка любит молоко.
Моей бабушке is correct- it is in the dative. This is because the verb нравиться is used with the dative case for whoever is 'doing the liking', while the object being liked is in nominative. Perhaps someone else can explain it better, but think of нравиться as "is pleasing...". So: "milk is pleasing TO my grandmother". Milk is actually the subject (kept in nominative case) while grandmother acts as an object (in the dative case). Sorry I cant articulate it better!
Not a native, but I think it's because своя has to refer back to the subject, and моя бабушка is the subject.
You could say Я люблю свою бабушку, because своя refers back to я. Or maybe Моя бабушка любит своё молоко, e.g., if she has her own particular brand of milk, or milks her own cows, and prefers her particular variety over others.