Why does automatic trim exist only in the pitch axis?
The pitch requires trim because pitch is related to airspeed and engine power. As both of these change quite frequently (except for the cruising phase of a flight), it is convenient to adjust the aircraft's "natural pitch" during climbs or descends, or after changing flaps.

Most aircraft* will not exhibit a tendency to roll / yaw to one side in flight, hence, it is already stable in these axis and trim is not required.

Automatic yaw trim does exist on some multi-engine aircraft. When one engine quits, the flight computer will automatically adjust rudder to compensate the undesirable yaw due to asymmetrical thrust. On an aircraft without such system, the pilot must manually use the rudder pedals, and the amount of rudder has to be adjusted as engine thrust or airspeed changes.

* Small single-engine airplanes do exhibit roll and yaw moments due to the torque and p-factor of the propeller, but these aircrafts are marketed for affordable price and easy handling, so auto-trim is not implemented.