This study introduces 2D Bell polynomials alongside Appell polynomials, examining their properties through generating functions. We delve into various characteristics of these polynomials, such as explicit forms, summation formulae, recurrence relations, and addition formulas. Additionally, we introduce the 2D Bell-Appell-based Stirling polynomials of the second kind, detailing their associated outcomes. This research aims to deepen the understanding of 2D Bell-based Appell and 2D Bell-Appell-based Stirling polynomials within mathematical analysis, with significant implications for approximation techniques. The findings could inform the development of new mathematical theories and applications, enhancing methods for approximation in various contexts.