<p>Microscopic insights into nonlinear interactions are essential for advancing polaritonic devices. Existing studies often rely on phenomenological models that overlook important many-body processes. Based on a material-specific and predictive approach, we investigate monolayer and homobilayer MoS₂ embedded in a Fabry-Pérot cavity to characterize the exchange, saturation, and dipole-dipole contributions to polariton-polariton interactions in these technologically promising materials. A key finding is that the exchange interaction induces asymmetric energy shifts of the lower and upper polariton branches in a detuned cavity, a behavior driven by the difference in their excitonic character. Furthermore, we demonstrate that temperature and electron-photon coupling determine the energy renormalization through the equilibrium polariton distribution. In homobilayers, the dipole-dipole interaction is mediated by the interlayer character, enabling electrical control and facilitating the electric-field-induced closing of anti-crossings due to dipolar-interaction shifts. The gained insights on polariton-polariton interactions are important for the development of ultra-compact polaritonic circuitry.</p>