We would like to replace our basement bathtub with a shower. The plummer told us, that this would not work, because a shower drain is bigger then a tub drain pipe. Is this correct? If yes, what would happen if we were to find a shower with a drain the size of our tubs, would that work?

The new revision in the sbcci code allows a 1 1/2" trap for a shower. i think it is bad, but now legal. call your local municipality and ask them what is allowed in your area.
or you can excavate the old trap and see if it is 2", or change it to 2"