I think part of the confusion here is that apparently the third person possessive pronouns are not declinable (https://en.wiktionary.org/wiki/jej#Polish). In other words "jej" is already the genitive (possessive) form of ona and doesn't match the noun like an adjective would. Literally it's "dog of hers."
I would advice to call them possesive, and not genitive. nouns in genitive when they indicate possesion, are alvays after the noun they describe, while pronouns are before. Also jej and ich are one of two possible genitive forms, and jego is one of three possible genitive forms. But there is ony one version of respective possessive pronouns. ( dla niej , dla jej psa/ do niego, do jego domu, nie widzę go, nie widzę jego psa/ do nich, do ich domu)
jej pies/ pies Kasi