I opened chests and in one was a mystic dragon and another a puppy dog which are both pets. They are..

So you press your animal icon and then at the bottom, press collection, once you press that, it will show up with your animal, and with icons on the left with skins, clothes, hats, actions and body size, now look on the right side and u should see ur icon, the pet icon and den icon. You wanna press the pet icon, once you press the pet, it should have your pet and a paw icon to customise its name and gender, below that should be the skin icon, then the hat icon, you wanna press the skin/mask icon then pick whatever pet you want!