No. f/2 is f/2 only in exposure triangle. f/2 is not f/2 when considering DoF or (total) amount of light per photo (not per pixel).
Think about this - aperture is the size of the lens opening. If you do not use the whole lens opening, the whole aperture size, that means that you are using smaller than maximum aperture. What does it matter to you if lens has a certain aperture when you are not using the light that that aperture passes through - you are using only a part of it? So, on crop sensor that f/2 aperture is effectively smaller than it would have been on full frame senzor. What is really important - physical size of an opening, or size of that opening that is actually used for taking a photo?
You have to realize that F-stop of f/2 (or F2) is a relative aperture - relative to the focal length! Without focal length you have nothing! On the other and, real aperture is what defines DoF and total amount of light per photo. In other words, when using 50 mm f/2 lens on a FF body, real aperture size is 25 mm. If you use f/2 on 100 mm lens, real aperture size is 50 mm. Wow! Mind blown! The same relative aperture (the same F-stop) equates to different real apertures on different focal length lenses!
But, if you use equivalent F-stops - 100 mm F4 on FF gives aperture size of 25 mm, which is the same as 50 mm F2 lens - real aperture size in both of those instances is 25 mm. F-stop (relative aperture size) is different, but real aperture size is the same. And we can expect the same FoV with that combination, the same DoF, and the same amount of light per photo (which means we should drop ISO on MFT camera by 2 stops - and obviously, FF cameras have about 2 stops better ISO performance - everything checks out).