Earlier quoted context omitted.
I don't know why people say this. When I wore glasses I cleaned them with my cotton shirts for over a decade and they didn't get scratched up, at all. I don't see how cotton would scratch glass to begin with.
They live in an imaginary world where no one ever cleaned glasses until microfiber cloths were widely available. To clean glasses safely you basically need a soft, clean cloth. Cotton is totally fine. You could get away with a soft clean sponge, too. Or even a soft-ish piece of paper (which is what most disposable lens words are.)
The latter does tend to scratch over time, if perhaps only slighly, but the damage can accumulate.
I'm on team soft-cotton, with a very-well-worn bandana serving as my usual cleaning material, plastic lenses, no scratches.
Another sin, for glasses, is laying them lens-down, or face-up, on surfaces when not in use. Lens-down of course grinds the lens into whatever is on the surface. Face-up, as you'd wear them, is vulnerable to flipping over (most glasses are top-heavy), so upside down is preferable. Or folded, with the earpieces down and lenses up. In a case is of course preferable to either.
Leaving glasses randomly on chairs, sofas, beds, etc., is also an invitation to catastrophe.
I've lived with people doing many of the above, and their glasses were perpetually scratched and damaged. Given the high cost of a new pair for many of them, this was ... curious.