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. Slots open after you finish the tutorial. You can't move a body from the first slot: you can only switch it with a different chassis. It is tied to the current game mechanic: the game won't allow a player to sell his last body.