Far field radiation and spherical waves
A plane wave means that the vectors of the electrical and the magnetic field as well as the poynting vector will be perpendicular to each other. This will be the case at any point in the far field. At each observation point it looks as if a plane wave (or a spherical wave with a very large radius) would arrive there from a great distance.

However, it does not mean that the intensity of the electromagnetic field will be the same at any point. With the exception of an isotropic antenna, the intensity of the wave will depend on the direction from antenna to the observation point. This is what is expressed by the beam pattern.