What rarity tiers exist in Aniimo?

Four tiers is a short ladder, so the labels do a lot of work. Common sits at the bottom and perfect at the top, with rare and elite between them. When you see an Aniimo described as elite, that is this scale talking, not its combat ability.

Rarity is also not the only thing being graded. Aniimo are split into five combat roles as well: attacker (DPS), healer, support, breaker, and regeneration. Role and rarity are independent, so a common can be a healer and an elite can be a breaker. Knowing a creature's tier tells you nothing about what it does in a fight.

Rarity tiers and potential grades are two different scales

Potential grades, sometimes called potential ratings, also come in four tiers: common, good, elite, and perfect. Three of those four names are shared with the rarity ladder, which makes the two easy to mix up. The tell is the second tier. Rarity's second tier is rare; potential's second tier is good. If you are reading a stat panel and see "good", you are looking at the potential grade.

Upgrading an elite Aniimo to perfect

Rarity is not permanently fixed. An elite Aniimo can be raised to perfect using a Capability Fruit, which grants plus one Initial Potential in that creature's recommended stat.

The catch is eligibility. Only elite Aniimo showing a sparkling potential aura can make the jump, so plenty of elites will never be candidates no matter how much material you hold. Check for the aura before you commit a Capability Fruit to a creature.

Not settled: The only rarity upgrade described in the game is elite to perfect. Whether common or rare Aniimo can be moved up the same way is not established.