This paper introduces and analyzes the approximation properties of bivariate sampling Kantorovich series in weighted spaces of functions. We define the operators using a kernel function that satisfies specific conditions, establishing their linearity and well-definiteness within the weighted spaces of functions. The main results include proving both pointwise convergence for weighted spaces of continuous functions and uniformly continuous functions. Furthermore, we provide a quantitative analysis of the approximation error, deriving an estimate for the rate of convergence in terms of the bivariate weighted modulus of continuity, which demonstrates the dependency of the error on the moments of the kernel function and the smoothness of the approximated function. The theoretical findings are further illustrated through numerical tables and graphical examples in the final section, providing a visual confirmation of the convergence properties and the accuracy of the approximation.