Good question!!! I wondered too, so I did some research.
I learned that the animal is a dog which St. Phillip obtained from a Cardinal in Rome.
"It is said: He had a dog and he would ask people to take his dog for a walk. But, as they started to walk the dog, he would stop them and say, “What are you doing? I mean carry him.” If you saw anyone in Rome carrying a dog you would know that Philip had asked him to take his dog for a walk. " (
peacepilgrim.fotopages.com/?entry=450372)
I admire St. Phillip Neri!!! He makes me happy to think about him and his character.
St. Phillip Neri was a real funny man, an all around funny saint. I would say that, to my knowledge, he is about the funniest of the well-known saints. He brough cheer to nearly everyone. He carried with him rays of sunshine.
He would carry a joke book in his pocket. He would walk around in public with a half-shaven beard. He would wear bizarre clothing.
“St. Philip Neri preferred spiritual mortification to physical mortification. When one man asked Philip if he could wear a hair shirt, Philip gave him permission – if he wore the hair shirt outside his clothes! The man obeyed and found humility in the jokes and name-calling he received.” (
Catholic.org)
"Humility was the most important virtue he tried to teach others and to learn himself… When one priest gave a beautiful sermon, Philip ordered him to give the same sermon six times in a row so people would think he only had one sermon. " (
Catholic.org)
Learn more about him at:
catholic.org/saints/saint.php?saint_id=97
“If one had to choose one saint who showed the humorous side of holiness that would Philip Neri.” 
(
Catholic.org)