The garage door is an example of constrained motion. End A is constrained to move in a horizontal line. Points B, D are constrained to move in a vertical line and to stay in a rigid position relative to A, C and C, E respectively.
The object dropped from a height is not constrained while it is freely falling.
A fairer comparison would be to examine the falling object when it touches the rigid floor. Now it is subject to a constraint, which prevents it from moving any lower. Usually the object is deformable like a ball. Its centre of mass moves lower but slows down from a velocity of $\sqrt{2gh}$ downward to zero over a short distance, usually much less than the radius of the ball, then reverses direction. Like the garage door, at the lower limit of its motion its vertical velocity is zero.
For the ball its gravitational PE of $mgh$ has been transformed first into the same amount of KE, then into the same amount of elastic PE as it deforms. Assuming there are no losses of energy due to friction, air resistance, hysteresis, etc then it will bounce back up to its original height.
For the garage door its gravitational PE when in the fully open position has also been transformed into the same amount of KE when the door reaches the lowest limit of its movement. The COM of the door is no longer moving vertically, but the two panels AC and CE are rotating about B and D respectively - so the door now has rotational KE. If there are no friction losses the door will - like the ball - swing back up to its initial height, its rotational KE being transformed back into linear KE and finally to gravitational PE.
