We describe the evolution of the SU(4) Kondo effect as the number of magnetic centers increases from one impurity to the two- dimensional (2D) lattice. We derive a Hubbard- Anderson model which describes a 2D array of atoms or molecules with twofold orbital degeneracy, acting as magnetic impurities and interacting with a metallic host. We calculate the differential conductance, observed typically in experiments of scanning tunneling spectroscopy, for different arrangements of impurities on a metallic surface: a single impurity, a periodic square lattice, and several sites of a rectangular cluster. Our results point towards the crucial importance of the orbital degeneracy and agree well with recent experiments in different systems of iron(II) phtalocyanine molecules deposited on top of Au(111) [N. Tsukahara et al., Phys. Rev. Lett. 106, 187201 (2011)], indicating an experimental realization of an artificial 2D SU(4) Kondo- lattice system.