Does the German noun Schwan “swan” somehow relate to the German noun Schwinge “wing”?

You are making some very tortured associations. "Etymology" means the study of the evolution of words, not finding accidental similarities of sounds. The human voice is capable of about 40 different sounds, and some sounds are preferred depending on one's culture. So it is inevitable that some words are going to sound alike even though they are not associated at all.

Consider the American word "shed". This word appears in four different languages with four meanings only vaguely similar. In Wales the noun means a place where a decision must be made, such as the part in one's hair where the hair must go either left or right, or a divided road where one must go one way or the other. In England it was a verb meaning to take off quickly, as water off a duck's feathers, or an animals winter fur when spring comes. In another language, a shed was a flat tilted roof without regard to whether it was enclosed, but in America it came to mean a three sided structure using a wall of another building as the fourth side, without regard to the manner of the roof. Also in America the word "watershed" was coined, meaning the catchment basin between two ridges, whereas the Welsh word "shed" meant the ridges.