is that ... a Pigeon ?

I think it's a Budgie

Aww... it's so cute.
But, why is the poor thing stuck on its back like that?

It's a budgie. It's possible that the owner put it on it's back for a trick, and it's not allowed to get up until it's owner says so for a treat. I think it's along the line of telling a dog to "play dead".