The real secret behind the magic eraser and all the "copies" (technically it's not a generic or fake eraser, it's just a lot less actual eraser on it) is Melamine resin/foam. On the generic or dollar store brands you get a 3/4 inch sponge and about a 1/8th inch melamine foam piece.
They do this because perception is that thin isn't useful and the sponge can be used anyway to wipe up stuff you don't want to waste the magic eraser on (because it gets used up pretty fast if your using it a lot). It's also a necessity because to effectively use the Magic Eraser you need to get a grip on it. The sponge also makes it easier not to press so hard- acting like a shock.
Soaking the eraser in cold to warm water expands the melamine foam fibers much like those toys
you soak in water and they grow, only this is more like a sponge.
The expansion is powerful because it creates spaces between the fibers of the melamine resin foam (the magic eraser part). Like a sponge, the water also tends to dissolve or loosen up the dirt or what you want to remove from the surface, but the Melamine foam can dig into the pores a much deeper clean.
With normal sponges dirt can often be dissolved and "absorbed" up by the sponge, but if the dirt or other ingredient can't easily be dissolved the magic eraser comes to the rescue with a longer, harder, more efficient scratcher that can dig deeper into the pores of your counter top (or any surface).
Dirt and other things get caught between the pores in the surface area and tend to fall too deep for a normal sponge to get to and extract.
This is what is known as either a hard to remove "stain" or just a 'stain' meaning we've given up trying to remove it.