Why can the polarization of an antenna array be determined by the polarization of its individual elements?
The polarization is defined by the direction of the electric field component. So since all the elements in the array would have the same electric field, the polarization for one element would be the polarization for all the elements. Electromagnetic fields abide by linear superposition so by varying the electric field in one direction and adding more electric fields in the same direction, the array's electric field direction stays the same. 

Thanks to @MarcusMüller for the help!