Assuming your assumption is correct, i.e. that riding in a car is the same as riding the subway.*
Can you switch drivers and have the Gentile drive your car and you be the passenger?
Possibly. There is a Mishna in Shabbat that you can give your valuables to a non-Jew to carry.
But... you have a problem with Techumim. I.e. if you're outside Techum Shabbat** of your destination then when you get out the car you will be limited to walking 4 Amoth. Since subways don't leave town, it isn't mentioned there.
I know that you cannot benefit from a Gentile doing melacha. However, here, the Gentile is driving your car for his own benefit, as he also wants to get to the same place as you do. You just happen to be the passenger, now. Is there a problem with his driving the car?
Assuming that he also needs to go to the same destination, then it sounds OK. Compare to the Mishna of the no-Jewish sailor making a ramp for the Rabbis to leave the boat.
But if he's going to detour to drop you off then it's probably forbidden.
Also add the problem of how to get out the car. Opening doors will possibly cause lights to go on, on the dashboard and the ceiling; you'd need to ask the non-Jew to let you out, possibly in an indirect manner. (
Um, how does one open this door? may not be allowed since one cannot mention the forbidden act in the hint .)
This again is not an issue in a subway.
Also, once he's arrived, can he turn off the car, as now, it seems that having the motor off seems to be for your benefit - it's your car, not his?
Well, you can't ask him to turn off the motor, but you don't have to stop him from doing so. Compare to the Halacha of a non-Jew who wants to extinguish a fire in your property. (As a general rule, extinguishing is easier than other Melachot, since it's a Melacha She'eino Tzricha L'Gufo.)
* Not sure the comparison is valid, since, for example, your weight is negligible in a subway, but significant in a car. You don't have the issue of opening doors, turning off engines and other potential trouble spots.
** In a nutshell: If you're 1Km outside the city limits when Shabbat starts (at dusk? nighfall? twilight? - that a topic for another question). City limits are defined rather generously; draw a square around the city that includes all lived-in building plus 70 Amoth around them. but in a car at 60 Km/h if you're 2 minutes away and cannot see the city, you're probably outside the techum.