We investigate how non-Hermitian random-matrix universality emerges in a physical Floquet system with localized leakage. As a concrete example, we consider the leaky quantum standard map (L-QSM) of the kicked rotor. Since the closed map exhibits circular orthogonal ensemble (COE) statistics, we compare its spectral properties with those of the truncated circular orthogonal ensemble (TCOE), which models localized leakage by removing columns from a COE matrix and thus provides the natural finite-dimensional random-matrix benchmark for the open system. The local spectral correlations of the L-QSM and TCOE show excellent agreement, evolving from the COE limit, recovered only when the truncation is smaller than one full column, to the universal statistics of the non-Hermitian symmetry class ${AI}^{†}$, reached when the number of truncated columns is $n_L\gtrsim 100$, independently of the matrix size. Their global spectral properties, however, retain differences. At intermediate leakage, the L-QSM and TCOE exhibit an accumulation of long-lived resonances near the unit circle, reflecting residual unitary structure inherited from the closed-system limit, while their densities of states remain far from the Ginibre circular law. The L-QSM additionally exhibits system-specific features absent from the TCOE: nonergodic states strongly coupled to the localized opening produce an excess of very short-lived resonances, while phase-space stickiness affects their population of long-lived resonances. Only at strong leakage does the global density of states of the L-QSM and TCOE approach the Ginibre circular law. Our results show that local bulk correlations reach non-Hermitian universality at weaker leakage than global spectral properties, which retain system-specific features over a broader range of openings.