You can't say that the engine or chassis is the determining factor. Take Toyotas for example. The 1ZZFE engine is in the Corolla, Matrix, and Celica. Those are certainly not the same cars. As for chassis, the Neon and PT Cruiser share a chassis, as does the Sequoia SUV with the Tundra Pickup. Clearly, none of these are the same.
Cars do get classified as 4-Door Sedan, 2-Door Coupe, 2-Seater, SUV, Pickup, Wagon, all that. But those are generalizations, and there is some crossover, like a 2-door and 4-door Corolla, for example.
I think the people behind a design are as key as anything else, if not more. A specific design team designed the Celica, the Mustang (whatever generation you have), a De Lorean. My feeling is, that's it. The people who create, design, what have you, a car, decide its identity.