I know that Bastion's main avian companion, who appears to be some variant of a goldfinch, is named Ganymede. But what about the other birds? Each of Bastion's Epic and Legendary skins (aside from the Overgrown skin) show a different species of bird.
Defense Matrix has a Blue Jay.
Omnic Crisis has a Cardinal.
Antique and Woodbot have two different colors of a Woodpecker.
Gearbot has a Pigeon.
Steambot has a Dove.
Normally, I wouldn't expect such a minor character to have different names for each of the suits they wear, but Blizzard has been known to be quite a stickler for the smaller details.
Best Answer
We don't know.
We can speculate all we want, but in reality there is no given name for any of those alternate birds.