A pivotal challenge posed by unconventional superconductors is to unravel how superconductivity emerges upon cooling from the generally complex normal state. Some of the most prominent unconventional superconductors are oxides: strontium titanate, strontium ruthenate, and the cuprates exhibit greatly different superconducting transition temperatures $T_c$, and although their respective superconducting pairing mechanisms remain unknown, they are thought to differ as well. Here we use nonlinear magnetic response, a probe that is uniquely sensitive to the superconducting precursor, to uncover remarkable universal behavior in these three distinct classes of superconductors. We find unusual exponential temperature dependence of the diamagnetic response above the transition temperature $T_c$, with a characteristic temperature scale that strongly varies with $T_c$. We correlate this scale with the sensitivity of $T_c$ to local stress, and show that it is influenced by intentionally-induced structural disorder. The universal behavior is therefore caused by intrinsic, self-organized structural inhomogeneity, inherent to the oxides perovskite-based structure. The prevalence of such inhomogeneity has far-reaching implications for the interpretation of electronic properties of perovskite-related oxides in general.