You can save up to three different car setups and switch between them freely at any time. To do so, tap on the 1, 2, 3 icons in the Garage.
Each spare part can only be used on one car. The slots open after you finish the tutorial. You can't move a body from the first slot: you can only swap it with a different chassis. This is due to the current game mechanic: the game doesn't allow you to sell your last body.