There's two compatible definitions for work and power to be used here. The one you are thinking about is $W=F\cdot \Delta x$, work is a force applied over a change in position(distance), and this $P=Fv$, power is a force applied at a velocity. There is, however, another definition, $W=\tau\cdot \Delta \theta$, work is a torque applied over a change in angle and thus $P=\tau \omega$, power is a torque applied at a given angular velocity. Sorry for the Greek letters, that's just how we roll! (yes, I went there)
It turns out these two definitions are exactly equivalent, if you break objects up into point masses. If you have a rotating object and you sum all of the forces on the point masses in all the directions they are traveling (each point is going in a different direction, since they're moving on a circle), you get the equations for work and power based on torque. If you look at torques, and look at points that are moving in a straight line (which means their radius is changing at the same time as their angles are changing), you can derive $W=F\cdot \Delta x$ and $P=Fv$ from the angular equations.
The only difference between them is that one form is very good at capturing what is happening when objects are moving linearly, while the other is very good at capturing what is happening when objects are moving circularly. If you are the kind of person who thinks in coordinate systems, forces are good in cartesian (x/y or x/y/z) coordinate systems, while torques work well in polar cordinates. Thus we can choose which definition of power to use based on what symmetries we see in the problem. We choose our approach based on what equations make the most annoying things cancel out and dissapear!
In this case, you are interested in power at the wheel. The wheel is rotating, which means the equations of motion are simpler to understand if we think in torques. You can think in forces, and get the right answer, it just takes more mental arithmetic to get there.
If we think in torques, we see what "power to the wheels" would have to mean. Your wheels are effectively applying a torque to the ground (and one step before that, your axels are applying a torque to the wheels). Your wheels are turning at some angular velocity (some number in rad/s if you're thinking like a physicist, revolutions per minute (RPM) if you're thinking like a car person). The power to the wheels is the product of these two things.
If we want to think about "power to the body of the car," as in the thing which opposes air friction, we would naturally want to think about this in a linear space. We would then wish to use forces and velocities to determine power. And, indeed, we can see that the forces applied to the contact patch of the wheels gets applied in the direction of a velocity, doing work. It's just a little trickier to see how it gets from the engine to that contact patch because all of the parts in that chain are rotating, making it a bit more difficult to think in forces.
And if we're feeling a bit masochistic, we can also solve the equations of motion for the entire car around an axis of rotation. With four wheels, the equations get a bit murky. However, if our car was a unicycle, this may be a very natural way of thinking about things. On a unicycle, it is very reasonable to think of the force of gravity pulling you down as a torque around the axel of the unicycle that needs to be balanced by a corresponding torque by your feet on the pedals going in the opposite direction.