I've been over the issue of pistol stabilizing braces before. Attaching one to a pistol is perfectly legal. It is also legal to use it as a shoulder stock, though that's not the intended purpose. The BATFE issued a very specific letter to clarify this.
Black Aces Tactical recently manufactured a shotgun with the brace, and they've been informed that this is impermissible. People are now under the impression that this affects all uses. That is not the case.
The answer is simple and short. In legal terms, a shotgun is not a pistol. They are two different things.
As per 18 U.S.C., § 921(A)(5), a shotgun is defined as,
a weapon designed or redesigned, made or remade, and intended to be fired from the shoulder and designed or redesigned and made or remade to use the energy of an explosive to fire through a smooth bore either a number of ball shot or a single projectile
A shotgun is a smoothbore weapon designed to be fired from the shoulder. It has its own set of legal restrictions.
This ruling prohibits the use of a pistol brace on a shotgun. It has no applicability to actual pistols.