Automatic Transmission shakes car when shifting from Parking mode to Drive mode

Edit: Bad torque converter can also cause the issue you described. But its easier to diagnose engine/trans mounts.

I can't imagine theres an actual issue with the transmission if you say it drives fine. I would suspect bad transmission/motor mounts. The rubber on the mounts will wear away and thats when you feel shaking, while you drive the engine/trans will rely on the weight of the drive train which is why the shaking would go away.

Its not too hard to check with a friend. Have someone watch the engine with the hood up, never stand directly in front or behind a car when it starts! With them watching the engine keep your foot on the break and shift the car into drive so the car doesn't go anywhere. If you see engine movement you know your mounts are bad. Also shift into reverse with your foot on the break to check for movement. Remember you don't want the car to go anywhere, you just want to watch to see if the engine is moving freely. It will be easy to see standing from a safe distance. Engine movement is bad.