Is the dead zone on a console controller built in to the controller hardware or is it managed by the console itself

Short answer: Either of those, but usually both.
Long answer:
The dead zone can be influenced by the physical moving part (i.e. the analog stick or steering wheel, or whatever your talking about) in the way that the receptors that read the movement of the moving part could be limited to certain extremities. Mostly however, the dead zone on any controller is controlled by the software (in this case either the operating system of your console, or more likely the game itself) for the simple reason of personal preference.