Modeling and predicting the mechanical response of multiphasic biological tissue to external mechanical loading challenges researchers since decades. The Theory of Porous Media provides a continuum mechanical framework to investigate the underlying physical mechanisms within the tissue, particularly the interactions between the deforming solid matrix and the free moving interstitial fluid. Our nonlinear poro-viscoelastic material model can capture the complex mechanical response of e.g. human brain tissue, but it also possesses a strong coupling between several material parameters. First, the model comprises two timescales, one controlled by the solid viscosity and one by the permeability. Second, the solid volumetric stress response governed by the first Lamé parameter affects the overall porous material response in conjunction with the deformation-dependent permeability. Therefore, it is challenging to find experimental protocols that provide all necessary information for reliable and unique parameter identification independent of the choice of initial parameters. Based on numerical experiments for three loading protocols (cyclic loading, compression relaxation and a combination thereof), we perform an exhaustive inverse parameter identification study using various sets of initial parameters and compare the results to the \emph{a priori} known correct parameter set from the numerical experiments. Our results show that the reaction force alone does not provide enough information to reliably calibrate our poro-viscoelastic model. Especially the strong coupling between porous and volumetric effects can lead to physically unreasonable deformation states, while still providing a good but non-unique fit on the reaction force. We conduct the same numerical study again including the lateral displacement of our specimen as an additional information into the objective function of the inverse parameter identification. Our results demonstrate that including the lateral displacement significantly increases the likelihood to obtain reliable, physically meaningful parameter sets. Finally, we validate our numerical insights with actual experimental data from human brain tissue. We highlight the importance to record the lateral displacement during the experiment and consider it for the inverse parameter identification to accurately capture the mechanical response of biphasic tissues.