The classical Biot's theory provides the foundation of a fully dynamic poroelasticity model describing the propagation of elastic waves in fluid-saturated media. Multiple network poroelastic theory (MPET) takes into account that the elastic matrix (solid) can be permeated by one or several ($n\ge1$) superimposed interacting single fluid networks of possibly different characteristics; hence the single network (classical Biot) model can be considered as a special case of the MPET model. We analyze the stability properties of the time-discrete systems arising from second-order implicit time stepping schemes applied to the variational formulation of the MPET model and prove an inf-sup condition with a constant that is independent of all model parameters. Moreover, we show that the fully discrete models obtained for a family of strongly conservative space discretizations are also uniformly stable with respect to the spatial discretization parameter. The norms in which these results hold are the basis for parameter-robust preconditioners.