Can I have the same car and driver for a whole week in Naples?

Yes, and it is the arrangement most people settle into once they have tried it. One chauffeur assigned for the stay rather than a car summoned each time.

What changes is not the vehicle. It is that by the second morning nobody is repeating the address, the bags go in the same way, the temperature is already right, and the driver knows you are the sort who leaves ten minutes early or the sort who does not. A week of that is a different experience from seven separate rides.

It is also simply easier to plan around. The car is held for your days rather than requested against them, so a dinner that runs long or a meeting that moves does not become a transport problem. Days can be set or left open, and the concierge holds the schedule rather than you.

For a longer stay in Naples the usual shape is a car for the working days and the evenings that matter, and nothing on the days you want to be left alone. Tell the concierge the week and they will put it together.