Not a fake but a cut square/cut corner from a letter card. You folded the pre-stamped form in half and it was gummed along the edges to seal. 3 sides were perforated so the receiver just tore off the margins along precut perfs to unfold the card and read the message. See if I can find a whole form to show this.
EDIT: Here's a different one, mint, folded:

Similar to some mailed forms used today