If you're curious to check out the names of the pets that are unique (so not different elements), I have it for you below:

Killarpillar
Rockbeast
Tiny Dino
Mollyball
Ghost Blade
Sneek
Spry
Mrau
Dreadlord
Thorny
Ruhk
Cuddly
Nightshade
Marid
Salamancer
Mini Rockbeast
Manticore
Fierce
Thermacore
Phoenix
Flare
Ifrit
Reiku
Gobby
Ancient Pact
Aethon
Illusory
Hinn
Kangaroot
Chocolate
Wild Palgrus
Wochowsen
Jann
Painguin
Pinata
Olympian
Aquamarine
Culutiwa
Shiqq