When I equip new items on my Aavegotchi, it's fun to see the overall rarity type increase as the rarity score increases. I just upgraded my first rare to mythical and see that the background is now pink when I click on this Aavegotchi.
However, the specific trait labels don't change. For example, my Aavegotchi has a base aggression of 92, which makes it "warlike". I equipped it with many items to increase aggression to (92 + 6) which should put it into the mythical class of "based". It would be satisfying to see this label change as you equip or unequip your Aavegotchi.