It's a kickdown switch. It forces a downshift in cars with automatic transmissions. Often when you floor the pedal the car will downshift, but not the to very lowest gear possible. Pressing that button tells the computer you want the lowest possible gear.

It does not activate overdrive. That wouldn't make sense. When you press the gas pedal all the way, you want to shift down a gear -- not up one.

AFAIK it has nothing to do with the stomp test, either. You can complete the stomp test without pressing the pedal far enough to trigger the button.

I don't know what it's for in a manual car. Unless your car used to be auto?