Having worked as a journeyman mechanic for 30 years, I have changed out hundreds of engines, transmissions, and differentials, in all manner of machinery and vehicles. The process is pretty much the same, be it a tractor, car, heavy truck or trailer mounted equipment.
Most important is good planning beforehand, selection of quality tooling, and having enough room and an organized work area.
Reasons for removing (and subsequently reinstalling) an engine or transmission separately might be that one requires repair or maintenance, while the other doesn't, or, when performing a full restoration of a car. Then it makes sense to join the engine and transmission before installing them in the car. I even dress the engine fully before dropping it in to reduce the amount of time spent bending over the fenders afterwards.
I lost count of how many times I have removed and replaced the engine in my race Z, but the last time was after the crankshaft broke. Nothing wrong with the transmission, so I rigged up a way to hold the front of it, and pulled the engine only.
Soon, I will be pulling the engine and transmission from my brown 72 Z. I will be pulling them together, as the engine needs a rebuild, and the transmission (an automatic) won't be going back in. And when I reinstall everything, it will most likely be as a unit.
So there isn't really a right or wrong way to remove and install these things, provided it is done safely.