An F-number is a ratio commonly thought of as being derived from the focal length of the lens divided by the diameter of the aperture.

More correctly, though, it is the ratio of the focal length of the lens divided by its entrance pupil, which, for most optics, is not quite the same thing. (The entrance pupil is the image of the aperture as seen through the front of the lens. Optical elements in front of the lens aperture typically magnify the apparent size of the aperture.)

This value gives you the (approximate) ability to get the same exposure using the same shutter speed for different lens focal lengths set to the same F-number. I say approximate because an F-number does not take into account transmission light losses due the optics themselves, so a simple prime lens set to f/2.8 may actually be noticeably brighter than a complex zoom lens with many lens elements also set to f/2.8.

(Cinema lenses often cite a T-stop number (eg. T2.8), which does take transmission losses into account. A T-stop is equivalent to an F-stop (or F-number), except the the lens's aperture settings are calibrated to match an ideal lens--one which transmits 100% of the light it receives.)

To answer your other questions, the actual focal length of your compact camera lens is much shorter than than the lens on your Canon, so the entrance pupil required to deliver the same exposure is correspondingly much smaller in diameter.

As for the differences in shutter speed when both cameras are set to the same f-stop, different cameras' meters expose to different standards, different sensors set to a particular ISO vary markedly in sensitivity, and of course, the lenses involved almost certainly have different levels of light transmission loss.