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Here we just have a phrase outside of any context, so we use the basic, Nominative form. We'd use the same form if "His dog" was the subject of the sentence (His dog is eating meat = Jego pies je mięso).
In the other sentence, "his dog" is the grammatical object. The verb "mieć" (to have) takes a direct object in Accusative. "jego psa" is the Accusative form of "jego pies".
(Most possessive pronouns undergo declension just like nouns do, but 'jego'='his', 'jej'='her' and 'ich'='their' do not, that's why it's "jego" in both exercises)