swallow

Where it comes from

Swallow ordinarily means to make food or liquid pass down the throat; separately, a swift-flying bird. English inherited two different Old English words. The bird was swealwe, with Germanic relatives such as Old High German swalawa. The action was swelgan, to swallow, related to Old High German swelgan. Their modern spelling converged, but dictionaries trace separate histories. The noun for a mouthful was later formed from the verb. Figurative uses such as swallowing an insult or believing a story extend the act of taking something in.

How the meaning developed

None of those senses explains the bird's name; the bird is an accidental homonym.

In use

The bird and the eating verb have the same spelling.
Did one meaning create the other?

Old English had one word for the bird and another for taking food or drink down. Their spellings later converged.
A bird may swallow food, but that act did not give the bird its name.