The myth appears to have spread through email forwarding in the 1990s, sometimes with the addition that nobody knows why. In 2003 researchers at the University of Salford recorded a Pekin duck named Daisy in an anechoic chamber and then in a reverberation chamber, and demonstrated a clear echo in the reverberant recording.
The reason the echo is not obvious lies in the structure of the call. A quack is relatively quiet, fairly long, and ends with a gradual decay rather than an abrupt stop. An echo returning during that decay blends into it, unlike a handclap where a sharp onset and sharp end make the reflection distinct. Ducks are also usually heard over open water and marshland, where there is little to reflect sound back.
The finding is a small demonstration of a general point about acoustics: whether a reflection is audible depends on the envelope of the original sound and on the listening environment, not on any special property of the source. No animal sound is exempt from reflection.