I wonder if the "A320 simulator training" you have seen is actually Multi Crew Cooperation / Jet Orientation training. I know it's quite commonly done on both the A320 and the B737 types.
If so, this is not training to actually fly the type mentioned, it is merely being used as a vessel for another element of training. To actually qualify to fly aircraft like the A320 / B737 you need to do a type rating course, which is fairly extensive and time consuming and will training you for a specific type or series of aircraft.