"هٰذا الْكَلْب كَلْبي."
Translation:This dog is my dog.
Nope. In كلب كلبي we have two nouns bounded together (following each other), and in such case we go into the realm of (Genitive) or as some call it (of-relationship). So, كلب كلبي would be understood as (dog of my dog).
Since you've mentioned it, actually deriving an adjective from the noun كلب (dog) is possible by adding the usual suffix for relationship which is also ـي (in Arabic we call it ياء النسبة, the Yá of relationship), and maybe new-comers would mix up between (my dog) كلبي (kalbí) and (canine) كلبِيّ (kalbiy). Notice though that in the adjective there is a little stress on (Y) at the end, while in the possessive case (my dog) the word ends in a long vowel (-í).
Also, the context would actually give a hint about the word in use of course.
I think Duolingo is doing a literal translation here.
I would translate This dog is mine more as هذا الكلب لي, where لي (lee) means for me/mine.
Of course This dog is mine is correct as well in meaning but as I said, Duolingo often works by literal translation (I'm guessing for educational purposes and training).