I think that a way to lessen saturation would be if we could pick a few details when ordering new animals.
I'd like to be able to choose eye colour, fur length and colour, leg length, tail length, number of spots (for dalmatians, particularly), mood disposition, you name it.
Then we should not have just one price per animal but instead a multitude of small items that would add up. And you would have to set a different price for each and every minor detail.