Relative to their width people on bikes move a lot more side-to-side than a car does - it's not uncommon to see a person on a bike weaving a couple of feet in either direction as they pedal. If cars did this the lanes would have to be 2-3 times the width of the car on either side to accomodate the movement.
Further, having one hand off the handlebars massively diminishes your ability to react safely and accurately.
Granted, the worst that might happen in a bike-on-bike crash is a broken bone or concussion, but for people who aren't at peak health (or who just get really unlucky) this could be a significant, even life-changing event.
