It's not a pet bank, it's your normal bank. A pet will go into it even if it is 'full' so you don't lose it. Remember that pets with long timers (or if you check immediately after hatching) will appear as eggs until they hatch. They won't show at all in your housing inventory (the pet section when you are placing objects down in your house) until they have hatched and can be placed.
Where would I find this bank at then and will it show me what I have in it ?
The bank in your house/dorm room. When you interact with it click the left hand tab that says 'bank' (it will start on your backpack and to the right gives you the shared bank - where you can pass items to your other wizards). It looks like a wooden chest and is usually inside the house near the main entrance. In your dorm it is just near the door leading out to Ravenwood.
I hope this helps and I hope you find your missing pets
So i hatched a pet with one of my friends then i transferred the egg to one of my wizards, while in battle the egg hatched. After the battle i got into my bag just to see it wasn't there, i checked my bank, my shared bank even my other wizards and it is nowhere to be seen
