This research investigates cosmic dynamics within the context of $f(\mathcal{R},\mathcal{L}_{m})$ gravity, concentrating on a binary mixing of perfect fluid and dark energy in a Plane Symmetric space-time. By incorporating the non-linear form of $f(\mathcal{R},\mathcal{L}_{m})$ as $f(\mathcal{R},\mathcal{L}_{m})=\frac{\mathcal{R}}{2}+\mathcal{L}_{m}^{\alpha }$ , it investigates late-time cosmic acceleration and the transition from matter-dominated to dark energy-dominated epochs. The analysis includes the quintessence and Chaplygin gas models, demonstrating their role in the dynamics of energy density, effective pressure, and anisotropy. The model is validated through parameterization using observational data, such as Hubble parameter datasets, which result in an excellent level of agreement with empirical findings. Advanced diagnostics, like the jerk, statefinder, and $Om$ diagnostics, show that $f(\mathcal{R},\mathcal{L}_{m})$ gravity is different from previous cosmological models. This lets us explain the expansion of the universe in terms of geometry. This study provides a strong basis for future research on modified gravity, anisotropic cosmological models, and the role of dark energy in the evolution of the universe.
