When light gets reflected from a dielectric surface (like the glass of your windshield), the two polarization components of the light don't get reflected by the same amount. The coefficients of reflections for both polarizations are called the Fresnel coefficients.
More details here:
According to these expressions, there is even an incidence angle where one of the polarization components is completely transmitted. This angle is called the Brewster angle.
For example, if the angle of incidence of light on the windshield is at the Brewster angle, only one polarization will be (partly) reflected. This means that this reflected beam has a well defined (theoretical) linear polarization.
If your glasses are oriented to filter this remaining polarization component, you will not see any reflection from the sun through your glasses.
When you tilt your head, you tilt the polarization axis and some light will pass through again. The intensity transmitted is given by the squared sine of the angle between the polarization of the light and the axis of your polarizer.
This is also why photographers sometimes use polarizing filters to take pictures—to enhance or reduce these reflections.
Edit to answer your edited question :
As you can see from the first WP link, there's a fairly large band of incidence angles for which one polarization is reflected much more than the other one. That alone could explain why so many object polarize so well the reflected light you see.
An additional factor I can think of, would be that for an object to reflect light towards your eyes, it has to be at a specific angle. Depending on the position of the sun in the sky and yours relative to the object, it is not surprising that a large amount of objects reflecting light towards you are somehow not too far from the Brewster angle.
Why not finding a flat piece of glass (a watch for instance ?) and experimenting for yourself. Don't burn your eyes with the sun though !