Start with the animal table:

Animal(animal_id PK, ... animal attributes)

No need for join tables or surrogate keys on these tables. The animal_id can serve as both the primary key and foreign key.

Dog(animal_id PK FK, ... dog only attributes) Cat(animal_id PK FK, ... cat only attributes)

No need for surrogate key on pet, and don't forget the owner key:

Pet(owner_id PK FK, animal_id PK FK)

This allows multiple owners of one animal. If you don't want that you need a unique constraint on animal_id.