What is the gray felt meterial I got with my Sony?

I am guessing you bought this used and the seller sent this along.

I have unpacked scores of brand new (and used) Sony 5400's and none included any grey felt.

No, I bought it brand new. The felt has a sticky peel off back and my guess is that it's dampening material.

Yes, that is likely what it is but I have never found that included in any Sony XA5400 box I have opened.

Maybe it is for the bottom of the player, to protect your furniture, not damping. It sounds strange to use felt damping with no instructions.

Put it over your ears.

I can't understand why it would be damping material. I don't think you can over damp so any damping material would be applied at the factory. Sony doesn't want you to crack the case and apply this (read void warranty) and it wouldn't be acceptable to apply it on the outer casing. Maybe it's a crazy factory worker who sprinkled bath salts on his cereal in the morning?