Yes, pretty much. They are both 4 cylinders and 1.6 liters. 

For the engine, the engine block, cylinder head, oil pan, oil pump, timing gear is the same between the Cooper and Cooper S. 

The crankshaft, rods, pistons, exhaust valves, and internal lubrication are different.

If "there are many differences internally" and "Externally, almost everything that bolts onto the engine is different", can you please tell us, if "The engine block of the Cooper and the Cooper S may be very similar", what is exactly similar to both of them? They are 4 cylinders and are sold by BMW????

The engine block of the Cooper and the Cooper S may be very similar, but there are many differences internally, including materials, lubrication, and compression. Externally, almost everything that bolts onto the engine is different.

To answer your question, you 'can' physically bolt on the Cooper S supercharger onto the Cooper engine, but you will need to make massive changes to make it work. 

I recommend you trade up to a Cooper S - it will be less expensive, easier, and more reliable...

Toyota 22R is a good replacement!

Does a 05 mini non supercharged have the ports to put a supercharged on it? If not what’s the difference?