The gun is a "K" frame, as was the M&P model it was derived from.
The barrel was shortened to increase the marketability. Or maybe the muzzle end was damaged. The .38 S&W barrel dimensions are close enough to .38 S&W Special to work for both. The barrel was not shortened from the breech end because it is a lot more work to machine new threads and that end is not thick enough to re-thread.
The cylinder was reamed out to fit the .38 Special cartridge. If the cylinder serial number matches the rest of the gun, you can be sure of that. Also, examination of the chambers in strong backlight will show reamer marks where the chambers were deepened. If the chambers are original .38 Special, it is a replacement cylinder.
The .38 S&W cartridge diameter is several thousandths of an inch larger than the .38 S&W Special. Firing the Special in the reamed out chamers will produce bulged, and probaly split brass. This is not necessarily dangerous but it does ruin the brass for reloading. Also, the WWII era M&P was made of mild, automotive grade steels, and lacks the strength and safety margin of more modern M&P guns. It is not rated for +P ammunition.