The source of the noise at any given moment is where the tire is (or has just caused) the strip to vibrate. That point is moving away from the observer about as fast as the car is moving away from the observer
So, the doppler effect is present since the point of friction is moving to a different point on the road, even if the road is stationary.
I'll add one thing: normally you hear Doppler effect with a nearly constant sound (emergency sirens.) Here the pitches are changing, but because our minds instantly recognize the 8-step Western scale we can pick out the frequency shift even though the frequency changes between the different notes may be the larger change.
If it is the tire that makes the noise, it is no surprise that there is a Doppler shift.
In theory you could create the same effect by setting off fireworks.
Unless you're slipping, the point of contact between the tire and the road is always standing still with respect to you
Relativity. Christian Doppler's equation for the change in frequency of a wave, for subsonic velocities, is approximated by three variables: the velocity of the receiver relative to the source, the velocity of waves in the medium and the emitted frequency [1]. The first term is symmetric between a source approaching the receiver and a receiver approaching a source.
(The actual equation measures the velocity of the source and the receiver relative to the medium. Here, too, the symmetry between a source moving towards a receiver, the latter un-moving relative to the medium, and a receiver moving towards a source, the latter un-moving relative to the medium, is preserved.)