Is that true for all models? Even the “cheap”, mass produced ones?

More or less only for the high end models. Basically you apply for a model through the dealership > they send it to Italy where they accept it or not. If they do not they will usually offer you a lesser model first as you said. If they do then you get your car but you also have to sign a contract with them that they always have 1st call if you ever want to sell your car (if you break that they will never sell you again).