There are several threads on this topic, but essentially the answer is that it is mandatory to use the article before Mio/mia/miei/mie. Only exception: for singular family members (mia mamma, mio nonno, mia sorella etc).
Un like A is or indefinite article or a cardinal number, in Italian we have to use the partitive articles del/dello/della/degli (for masculin, feminine and plural)= some when we don't specify the quantity or quality of fish.
I don't quite get the possessive adjectives. Is it like saying "the dogs that are mine eat fish"? Is it like saying in Spanish "los perros mios come pescado"? And the only difference is that there is no word for "mis" when I say "mis perros comen pescado"?