I think the issue is a bird pet would have to constantly be flapping its wings to stay nearby which isn't so bad or unnatural as the flying pixies & dragonfly pet.
If it were to sit on your shoulder, it would be more of a memento because of how it interacts w/the player's character. i.e. the bird call raven landing on your arm.
Humming bird pet? Yes.
Parrot pet? Sure, but not likely to be able to land on your shoulder w/out being a memento.