We study the local limits of uniform high genus bipartite maps with prescribed face degrees. We prove the convergence toward a family of infinite maps of the plane, the q-IBPMs, which exhibit both a spatial Markov property and a hyperbolic behaviour. Therefore, we observe a similar local behaviour for a wide class of models of random high genus maps which can be seen as a result of universality. Our results cover all the regimes where the expected degree of the root face remains finite in the limit. This follows a work by the same authors on high genus triangulations.