How does it increase it by a factor of 4?
**Nerd alert warning**
If the triggers were pulled at exactly the same time, you double everything coming out of barrels...gas, unburnt powder, shot, and the velocity also doubles (not for each barrel, for the total sum).
Two commonly accepted equations for shotgun recoil are
Vg =(Ws + Ww + 1.5Wp) Vm / 7000 Wg
Eg =( Wg * Vg^2 ) / 2g
Where
g = gun
s = shot in grains
w= wad in grains
p= powder in grains
E= energy in foot pounds
m= muzzle
V= velocity in f.p.s.
W= weight
So, if you run those equations for a 3 inch, 12 gauge shell with a 1 3/4 shot charge, you'll get an approximate recoil of about 55 ft-lbs. If you double the velocity though, you'll see in the equation that results in a factor of 4, so the total recoil is actually about 220 ft-lbs.
Also, since it's twice as heavy, wouldn't that help a bit with how it feels (even though the force is more, the acceleration should be the same maybe?)
Yes, it will help with how it feels on your shoulder, but that only changes perceived recoil, the total is still roughly the same.
So, I definitely would not be a fan of holding that thing by the pistol grip and doubling the shotgun. That would suck.