Red velvety thing in nail/hair travel kit?

I have a hair/nail grooming kit that has this weird thing in it. It doesn't open all the way and is a little taller than the length of my palm. I remember my grandparents had something similar but it looked like a hairbrush. I thought it could be a shoe shining device, but since it doesn't open all the way that can't be right. I've searched for 'red velvet nail buff' and 'red velvet thing for nails' and only get results for the sand-textured block buffers and different red nail polishes. Anyone know what this is called and its proper use?