If we're talking strictly mammalian/real species, I always think back to Disney's "Robin Hood" for ideas!

Oh yeah, that is a good one.
There you had lions for kings
foxes as rebels
rabbits as peasantry
badger for monks
wolves/rhinos/elephants as soldiers
a snake as royal advisor

I would want any species to achieve any status but I would guess realistically, stronger pred species would hold authority positions and prey would be peasants and workers, depending on how dark you want the story to be they could even be slaves and food

To give it more context, I'm having this small game idea of someone trying to balance a kingdom and each representative is a different type of anthro animal.

Traders: Any species known for migrating. Birds and herd mammals mostly.

Nobles: Most would be "purebreds" -- mostly domestic cats and dogs -- who claim to have birthrights. The rest would be members of the merchant or peasant classes who excelled enough to marry into or inherit a title.

Peasants: Depends on the trade, but the simplest way to narrow it down would be herbivores in the fields (with each species having their own niche harvest) and carnivores in the forest (both as hunters and woodcutters), with all the various butcher/baker/candlestick maker jobs having descended from one of those two.

Church: Pigs. Because they're not kosher.

Foxes, Fennecs, Jackals, many Feline species could all be traders, merchants, or otherwise a seller of goods.

Blacksmith's could be anything from Canines (domestic), bovines, equines

Soldiers can be ANY species because an Army always needs recruits of any kind.

Monks: Mole's, Axolotl, Pangolin's.

Honestly there's so many things you could do with it and honestly it really isn't LIMITED by species, but more so by personality and if you want to build more character they would fall under different Archetypes.

Because ANY species also has potential for just being Peasants as well.