We report on the design, fabrication and characterization of a 64 pixel metallic magnetic calorimeter array that is read out by an integrated, on-chip microwave SQUID multiplexer. Based on the results of our comprehensive device characterization we refined the state-of-the-art multiplexer model which assumes each associated non-hysteretic rf-SQUID to purely behave as a flux-dependent inductor. In particular, we include the capacitance and the subgap resistance of the Josephson junction as well as screening effects and parasitic mutual couplings between different coils that show up only when a superconducting flux transformer is attached to the SQUID input. Thanks to these modifications, we are able to explain the occurrence of a magnetic flux dependence of the internal quality factor of the microwave resonators as well as to accurately calculate the characteristic multiplexer parameters. When combining the refined multiplexer model with the thermodynamical description of a metallic magnetic calorimeter, we find a reasonable agreement between our measurements and predictions.