I believe the button is for mobile controls, as you are also able to steer by clicking and dragging mouse on the screen. Why this mobile version was never released and why these controls are on the PC version confuses me as well.

Anyways, playing around with it you can also add more jump inputs by pressing the '5' and '+' keys on the numpad (with Window's MouseKeys enabled) with your mouse positioned on the jump button.

I wonder how far this can go, is a 'quad jump' possible? Although I doubt it's usefulness as even with 'triple jump' I found myself clipping through ground when falling back down in some cases.