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).
Not necessarily. Yes, there is size of photosites you don't want to go below, but the main reason is that MFT sensors in general receive less light per photo. Consider this - if you want to take a photo in low light, you might use 35mm f/1.4 on FF. To get the equivalent photo on MFT, you need 17,5mm f/0.7 lens. Does that kind of lens actually exist? If you use nifty fifty on FF (50mm f/1.8), to give the same amount of light to MFT sensor you would have to use 25mm f/0.9 lens. Does that lens even exist? Do MFT users use lenses that are that fast? No? No. They usually just use lenses with the equivalent focal lenght, but the same F-stop (which means 2-stop less light, and therefor get about 2-stop more noise at the same ISO).
So, that is the answer why MFT struggles with low light - because of the sensor size, and not having adequately fast lenses which would compensate for that small sensor size. For MFT cameras not to struggle with low light, used lens must have about 2-stops better F-stop than FF camera.