The story
Obscure. Speculative. Panhandling always seems to involve a container for receiving loose change, so perhaps the term refers to a small handled pan, or to the container as the pan and the arm(s) holding it as the handle. Alternatively, possibly from the notion that the shape of one's outstretched arm in relation to a seated or squatting beggar brings to mind the image of a panhandle.
Etymology from Wiktionary, CC BY-SA 4.0.