Purpose of the tiny cloth??

Yes, it's for protection versus cleaning. I recently saw an ebay listing where the seller of a fake Chanel wallet wrote, "Comes with cleaning cloth; don't use leather cleaner!"
 
The piece of felt I got was under the flap between the closure and and body of the purse. I thought to protect from any metal scratching on the CC?

Yes, I had this too but i also had a tiny cloth that was folded into the lipstick compartment. So it's just to retain the shape, not a cleaning cloth?