Cd is the coefficient of drag. It doesn't determine the amount of drag experienced by a car, that is determined by CdA, the coefficient of drag multiplied by the frontal area. So when you test in a wind tunnel, what you determine from measuring the force on the balance is CdA, and to separate this into Cd and A, you need to know the frontal area, which I guess is easily enough known from the CAD model for a car. I'm not sure why the convention with cars is to quote Cd rather than CdA. In cycling, for example, nobody talks about their Cd, only their CdA.
So yes, you can think of Cd as being determined by the shape, independently of the size, with the drag of an object depending on both Cd and size. It isn't quite as simple as that, as Cd can depend on the Reynolds number, e.g. a sphere has a Cd that significantly varies with Reynolds number:
https://www.grc.nasa.gov/www/k-12/airplane/dragsphere.html