This study proposes a mixed local discontinuous Galerkin (LDG) scheme for capturing shock-driven interfacial instabilities in compressible multicomponent flows. The proposed scheme is validated through canonical test cases, including sod shock tube, shock-driven light/heavy cylinders, and chevron-shaped, and shows excellent agreement with experimental and numerical benchmarks. It effectively captures key mechanisms including shock refraction, baroclinic vorticity generation, vortex roll‐up, and nonlinear interface evolution. A systematic parametric analysis is carried out for the single‐mode heavy fluid layer configuration to examine the impact of interface thickness on shock-driven instability development. The results reveal that thinner interfaces intensify baroclinic vorticity deposition and accelerate the transition to nonlinear stages, whereas thicker interfaces delay instability growth and mitigate small‐scale structure formation. Spatially-integrated measures of baroclinic vorticity and enstrophy further confirm that geometric confinement plays a pivotal role in regulating vortex evolution and interfacial mixing.