So the part assembly is called "Transmission Shifter" part number 05098490AA.

You do not need to remove the transmission to replace the Shifter assembly.

The linkage rod going from the shifter **** to the back of the transmission should not be able to fall onto the ground. Look at the assembly picture below. There is a support bar that would stop that rod from falling to the ground. Now if you were able to reattach the rod to the transmission running below that support maybe that could happen.

There is a mounting pin that holds that shifter assembly to the transmission. It has a retainer clip that attaches to the transmission case. Was that on, or did that come loose?