As identified above, this is actually cut from a letter card, the only one produced by the US (though not rare). It was a stationery type also used elsewhere. You will find from your "cut square" that the paper it is on is actually light cardboard. The whole thing:
You wrote your letter on the back, folded it and sealed it at the edges; it was gummed at the sides and along the flap. The perforations at the edges were torn off by the recipient to read the letter.