How close is the air resistance or drag force to being proportional to the airspeed squared?
Subsonic aircraft drag is composed of two major components:


Pressure drag. Most of this is induced drag, but also some is from viscous effects which result in less pressure on the rear sections of aircraft.
Friction drag from boundary layer effects.


At high speed, friction can be the dominant drag contributor and here we have a less than quadratic dependency because with increasing speed the influence of friction is diminished. This is expressed by the Reynolds number and for first order approximations the friction drag coefficient changes in proportion to $Re^{-0.2}$. Since Re increases linearly with speed, the friction drag force is approximately proportional to $v^{1.8}$ at high speed.

At low speed induced drag is dominant and this decreases with speed squared. Of course, there is still a contribution from separation (which is a major problem at low speed), so the proportionality is smaller. Still, at low speed aircraft drag drops over speed.

With cars things are simpler. Here we have a rather blunt body with lots of separation at the backwards-facing surface. The suction on that separated flow is the dominant contributor to a car's aerodynamic drag and is much less influenced by Reynolds number effects. However, a small decrease in the friction drag coefficient over speed on the surfaces with attached flow remains, so the proportionality is a whisker below 2. For engineering, using $v^2$ is good enough, however.