There's a decent amount of linguistic evidence that swan and song both come from an ancient root *swen-, '(make a) sound'. In Old English the bird was called swan, from a Germanic root *swanaz, 'singing bird', related to Old English geswin, 'song' and swinsain, 'to make a musical sound'.
This root *swen- is probably also the origin of sound, Latin sonus, Sanskrit svan, Irish seinn, 'play music'.
But, I hear you cry, the kind of swan we are most familiar with is the Mute Swan, which as its name implies doesn't make a very conspicuous or musical sound. How can calling it the singing bird make sense?
One theory runs like this. When the swan was dubbed 'singing bird' the people that named it were more familiar with a different kind of swan, such a Whooper Swan, which does produce a very noticeable sound.
Then as people and the Indo-European languages migrated south they met swans that didn't sing. But people were still vaguely aware that swan was something to do with singing, which produced a cognitive dissonance, which was resolved by inventing a myth.
The swan is silent all its life but then at the approach of death sings for the first and last time, its Swansong.
And so a meme was born.