The Mott metal-insulator transition, a paradigm of strong electron-electron correlations, has been considered as a source of intriguing phenomena. Despite its importance for a wide range of materials, fundamental aspects of the transition, such as its universal properties, are still under debate. We report detailed measurements of relative length changes ΔL/L as a function of continuously controlled helium-gas pressure P for the organic conductor κ-(BEDT-TTF)₂Cu[N(CN)₂]Cl across the pressure-induced Mott transition. We observe strongly nonlinear variations of ΔL/L with pressure around the Mott critical endpoint, highlighting a breakdown of Hooke's law of elasticity. We assign these nonlinear strain-stress relations to an intimate, nonperturbative coupling of the critical electronic system to the lattice degrees of freedom. Our results are fully consistent with mean-field criticality, predicted for electrons in a compressible lattice with finite shear moduli. We argue that the Mott transition for all systems that are amenable to pressure tuning shows the universal properties of an isostructural solid-solid transition.