That's a bit of a generalization.
Some folding stocks are fine on SGs, including with full power 12 gauge shot or slugs. I have an Advanced Technology folding stock on a Mossberg 500 that allows the SG to be easily stored, but deploys easily and is quite stable and comfortable in use.
Just don't believe the TV-Hollywood crap portrayed about short grip or pistol grip SGs. Nobody who uses or intends to use a SG seriously will have anything but a full or extendable stock on it.
Pump, bolt, and lever action guns are specifically exempted from the SAW (semiautomatic assault weapon) laws. The exception this is that revolving cylinder shotguns like the Streetsweeper fall under the law. You can put anything you want on an 870. The law also pertains only to rifles and pistols which have detachable magazines. If you were so inclined you could legally do up an M1 Garand in full assault weapon form. Stay safe, Gary
For the definition of SAW please see (a)(30) at http://www4.law.cornell.edu/uscode/18/921.html
For exemptions for pump, lever, and bolt actions see (v)(3)(B)(i) at http://www4.law.cornell.edu/uscode/18/922.html