High quality scissors, even with constant use, can easily last that long or even longer. Source: I used to work as a tailor. In fact many people in that industry prefer to use older German-made scissors as they are in many ways better than the stuff you can buy now.
He’s almost certainly referring to honing the blades. Actual sharpening would require equipment that would be just ridiculous to have around a hair salon.
A stone? A strop with polishing compounds?
Not necessarily. You can sharpen a knife(and I assume a scissor) with a sharpening steel. Which won't remove any material just straighten the edge.
Eventually that folds over after becoming brittle.
Honing is different than sharpening. Sharpening removes material, like a pencil. Honing merely aligns material.
A honing steel doesn't remove material, a sharpening steel does.