The use of electrochemical battery models, specifically the Doyle-Fuller-Newman (DFN) model, has gained significant traction in recent years mostly owed to the more idespread adoption of lithium-ion batteries in transportation, energy storage, and consumer electronics. Their utilization ranges from academic research to industrial applications; however, the parametrization of said models is tedious and poses a significant bottleneck for broader deployment. For this reason, the development of data-driven parametrization strategies has developed into a highly relevant research area in lithium-ion battery research. In this context, sensitivity analysis and identifiability analysis are increasingly employed to tackle questions related to relative parametric importance and the informativity of data with regard to certain parameters, respectively. However, the terms sensitivity and identifiability are often conflated in literature due to the underlying assumption that highly sensitive parameters ought to be identifiable. We therefore aim to provide a more exhaustive picture of the interrelation of parametric sensitivity and identifiability for the DFN model in the case of impedance simulation. To this end, we leverage algorithms for the efficient computation of generalized Sobol' indices via spectral representations and generalized polynomial chaos expansion-based surrogate models. Additionally, we employ neural posterior estimation, a simulation-based inference approach for efficient determination of the posterior distribution. This allows us to perform sensitivity and identifiability analysis in tandem from the same set of input-output samples. We believe that this work will contribute to a more thorough understanding of the intricacies of the DFN model, hopefully translating into advances in battery model parameter estimation workflows. Moreover, our approach for the simultaneous determination of generalized Sobol' indices and a Bayesian posterior distribution from the same data can be reproduced by researchers from various fields, thereby enabling insightful analysis for simulation models of all disciplines.