We perform a detailed study of the connection between hidden momentum and the Abraham-Minkowski debate about the electromagnetic momentum density in material media. The results of a previous work on the subject [P. L. Saldanha and J. S. Oliveira Filho, Phys. Rev. A 95, 043804 (2017)] are extended to the continuous medium limit, where some subtleties arise. We consider a polarized and magnetized continuous medium with applied electric and magnetic static fields, where the medium polarization can be due to either an electric charge density or a hypothetical magnetic current density and the medium magnetization can be due to either an electric current density or a hypothetical magnetic charge density. Each model leads to a different expression for the system material hidden momentum and for the electromagnetic momentum density in the medium. We show that the main results of the cited reference are sustained in the continuous medium limit: Abraham momentum is compatible with a model for the medium where the polarization is due to electric charges and the magnetization is due to magnetic charges, Minkowski momentum is compatible to a model where the polarization is due to magnetic currents and the magnetization is due to electric currents, and the expression $varepsilon_0mathbf{E}timesmathbf{B}$ is compatible with a model where the polarization is due to electric charges and the magnetization is due to electric currents, which is the natural model. These results are illustrated with the example of a uniformly polarized and magnetized sphere.