Much like the AR - order of assembly is important.
With the device the OP mentions there is no "assembly," it's simply held by friction and the shooters grip. Since there is no fastener and it readily falls off when released, it's not "attached."
The 1911 same as the Luger had a mechanical interlock which holds it in place.
Lets take this another step - No, I'm not writing any letters. If the buffer tube of an AR is not considered a stock by the ATF - who will classify any vertical flat surface attached to the rear of it as a "stock," then why not wear the flat vertical surface on the shoulder with a simple cup receiver to position the tube?
1) it's not attached to the tube, it's attached to the shooter.
2) recoil pads are part and parcel of "sporting uses."
3) It uses a harness shoulder pad instead of a shooter trying to place a buttstock to one side or the other
4) It can be devised to rest just above the actual shoulder pocket where the inline design of the AR puts it, reducing the amount a shooter needs to crink his head down to the sight line. Lots of AR shooters place just the toe of a normal stock in the pocket.
It would be something akin to a kydex butt plate but not directly attached to the gun.
Thats the sort of things that the law are usually never written to deal with, same as the '34 NFA never considering the mechanical innovation of the AR with it's separate upper and lower. It then causes legalistic interpretations based on case law to be made and those who do so know that concepts dreamed up without substantiation in the law will be directly challenged. Add 80 years of political wrangling and interpretations, along with court decisions which reinforce or undercut certain assumptions, and we get what we have now.
The '34 NFA started all this by attempting to ban handgun ownership completely. What got passed was a partial attempt to control misuse by legislating technology. If there is anything we should know for a fact today, it's that whatever technology a firearm exhibits has nothing to do with the intent of the user. That's why these laws are duplicative and unnecessary. It makes no difference whether a pistol has a stock or not - same as a knife having a push button to open the blade vs merely pushing the blade itself. The feature in and of itself isn't "moral," it's the users intent as expressed in what they do with it.
I carry a switchblade daily at work to open boxes, and it's a fine feature to have. It doesn't mean it's an evil or illegal device whatsoever - unless it's misused for some purpose that intrudes on the constitutionally protected freedoms of others. Same with a stock on a pistol - which certainly doesn't make it more concealable and only offers an incremental improvement in accuracy. We are talking handguns, which are in practical terms about 1/2 the power to consistently stop an attack when needed, and commonly used at close range - so close it becomes a question of "how did you miss?"
If stocks were made legal for AR pistols (which they were at one time) or any pistol, would the country become immediately endangered with shootouts in the streets and blood running in the gutters? Some would assert that we are already there - but the point is that the addition of a shoulder rest to a low powered limited range handgun isn't going to abruptly make that weapon a "game changer." The real issue is that it's about controlling the free and independent people of our nation - using privilege and financial obstacles as a gateway to ownership as proof of their character.
Since when is that High Road or even Constitutional?