Though the term came to us from German, the idiom exists because of a Greek legend - it was thought that certain swans sing a beautiful, mournful song right before they die.
It's a beautiful story, and it paints a lovely picture in my head. The only problem is that swans don't sing. I know I'm being too literal here, but swans honk. They grunt, they hiss, they moan and make all manner of sounds that aren't pleasant to hear. So the beautiful, mournful dirge that a swan might sing itself out with would sound a lot like rush hour traffic.