Singing in the shower also originates from the special acoustic characterstics that bathrooms and showers have.
As you've probably seen on MTV many times, vocal recording booths always have some sort of acoustic isolation material on their walls. These isolators prevent any echoes that may have been returned otherwise, echoes that sound incorrect or just plain bad.
Much like these vocal booths, showers also have a great acoustic isolation - steam. The little drops of water in the air absorb the echoes from your singing, and make it sound a bit better. This is a kind of real time sound processing. In the same way that electric guitars sound much better with a reverb effect on them, your voice sounds better in the acoustical context of the shower room.