We present a systematic study regarding the influence of surface charge doping on the magnetic anisotropy (MA) of capping 3d-5d/(4d) multilayers deposited on highly polarizable substrates such as Pt or Pd. The effects on the MA are investigated within a first-principles approach with a focus on Fe-Pt and Fe-Pd multilayers. For both layered systems, it is found that the MA is strongly altered upon charge doping, exhibiting different behaviour, however. A remarkable enhancement on the MA along with magnetization reversal is observed in Fe-Pt multilayers. In contrast, Fe-Pd multilayers have small MA energies as a consequence of their reduced orbital moments. Some MA features observed in the multilayers can be related with variations in the spin and orbital moment of the magnetic layers. Insights of the origin of the MA behaviour are investigated from a local perspective by analysing the d-orbital resolved local density of states in the framework of the second-order perturbation theory. Finally, the relation between the MA and orbital moments is discussed in terms of Bruno\'s formula.