How is the amino group basic?
Polarity and elementary electrostatics have nothing to do with acid-base definitions. Sometimes we try to find a reason, using our current understanding, and that ensues the danger of over-rationalization. At a truly fundamental level perhaps no chemist can tell you why an amine can accept another proton, may be someone can do quantum mechanical calculations and say why. So avoid over-rationalization... accept some observations as facts (temporarily until we develop a full understanding). 

One can titrate an amine with an acid, by this experimentally observed fact, exactly just like we titrate NaOH with HCl. So an amine group is indeed a base or say it behaves like base. The acidic proton cannot annihilate itself during a titration, it has to go somewhere. The amine group gets protonated, in return.