In this study, we examined a (2+1)-dimensional generalized breaking soliton system (GBSS) using both analytical and numerical methods. By applying a generalized direct algebraic method, we derived exact solutions that displayed a variety of solitary and periodic wave patterns. These solutions illuminated the interplay between nonlinearity and dispersion in several physical contexts, including fluid dynamics, plasma physics, and nonlinear optics. In addition, we developed a robust numerical scheme employing an adaptive moving mesh technique based on the MMPDE5 framework. Stability and error analyses confirmed that this method concentrated grid points around steep gradients, achieved second-order spatial convergence, and enhanced computational efficiency. By comparing numerical and exact solutions, we provided more profound insights into GBSS dynamics and facilitated future investigations of complex, multidimensional, and nonlinear wave phenomena.