God uses avian imagery self-referentially enough to make you notice.
We are not birds.
Obviously reptiles are out. Mammals seem a more intuitive trope, and we have the Lion of Judah and the Lamb of God. Maybe the struggles of our ancient fathers precluded further expansion on that theme. I don't think that's the only reason, though. The idea of being hidden in the shadow of His wings, being covered with His feathers, is curiously comforting.