There has been increasing interest on summary-free versions of approximate Bayesian computation (ABC), which replace distances among summaries with discrepancies between the whole empirical distributions of the observed data and the synthetic samples generated under the proposed parameter values. The success of these solutions has motivated theoretical studies on the concentration properties of the induced posteriors. However, current results are often specific to the selected discrepancy, and mostly rely on existence arguments which are typically difficult to verify and provide bounds not readily interpretable. We address these issues via a novel bridge between the concept of Rademacher complexity and recent concentration theory for discrepancy-based ABC. This perspective yields a unified and interpretable theoretical framework that relates the concentration of ABC posteriors to the behavior of the Rademacher complexity associated to the chosen discrepancy in the broad class of integral probability semimetrics. This class extends summary-based ABC, and includes the widely-implemented Wasserstein distance and maximum mean discrepancy (MMD), which admit interpretable bounds for the corresponding Rademacher complexity along with constructive sufficient conditions for the existence of such bounds. Therefore, this unique bridge crucially contributes towards an improved understanding of ABC, as further clarified through a focus of this theory on the MMD setting and via an illustrative simulation.