I think it's possible that this poem has a biblical allusion when it refers to the bird. Dickinson was raised in the transcendentalist era, and there was a lot of criticism oriented around the bible. The allusion could be that Jesus was said to give bread to people of poverty. He then told them not to eat the bread without giving the crumbs to the bird. I think this relates to the poem because man and nature was seen as one, while now they are considered to be vastly divided.
I'm not religious so I have no idea if this is correct or not. If there are any disagreements feel free to comment.