Couple of points:
1. Janus wasn't a Greek god, he was purely Roman. He was the God of beginnings and doorways, and in their pantheon was the first god.
2. The Romans didn't always have animal companions for their gods, thusly Janus had no animal as a favorite.
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I have never heard of a Dionisio, but Dionysus is a Greek god. He is the god of wine, his animal is the tiger, and his symbol is grapes.
The tiger was a sacred animal to Dionysus, God of Partying and Wine.
He is Greek god, but he does have a roman version Jupiter
Pan is god of shepherds and flocks (sheep/goats), Hermes of herds and animal husbandry, and Artemis those that are wild animals.
the greek god hermes the god of messages