An assault rifle is a rifle capable of both semi-automatic fire and full automatic and/or burst fire that uses an intermediate powered rifle cartridge. The S&W M&P-10 uses a full power rifle cartridge and can only be fired in semi auto.
The big Smith will work fine for home defense. It'll be a bit big, but not much bigger than a shotgun. Choose your ammo carefully to eliminate over penetration.
However, a 5.56 AR carbine is a better choice. Low recoil for fast follow up shots, good capacity, easy to use, light weight and manuverable- all advantages over a shotgun. The AR will also have less problems with over penetration through drywall than slugs or shot large enough to be effective for defense.
The M&P-10 will make a good hunting rifle. I mounted a 2.5x10 Nighforce on mine and I like it very much. DPMS has finally stepped up their game by using forged 7075 instead 6 series billet and have taken steps to lighten their rifles. Still, I like what I'm getting with my Big Smith- ambidextrous controls, R5 1:10 rifling and a good usable barrel profile that's 18 inches instead of 16. M&P-10s can be found for bargain prices and I think they're about the best value for a 308 AR, even with it's clunky flash hider