See my explanation here.
The pronoun has a 'front part' and an 'ending'. The 'front part' is chosen based on who the thing belongs to: sein- means the thing belongs to a male/neuter person/thing. The 'ending' is chosen based on the gender and case of the thing itself: -en is required for masculine nouns (such as Hund) in accusative case.
Other possible sentences to illustrate this:
Sie mag seinen Hund = "She likes his dog"
Sie mag seine Katze = "She likes his cat"
Sie mag ihren Hund = "She likes her dog"
Sie mag ihre Katze = "She likes her cat"
Er mag seinen Hund = "He likes his dog"
Er mag seine Katze = "He likes his cat"
Er mag ihren Hund = "He likes her dog"
Er mag ihre Katze = "He likes her cat"
Ihr mögt unsere Kuchen = "You like our cakes"
Don't be confused by the fact that the pronoun doesn't match the subject - in German as in English that can be ambiguous if they match, but they obviously don't have to:
"He likes her cat" = He likes the cat belonging to some other woman.
"He likes his dog" = He likes his own dog, or he likes the dog belonging to some other man.