<p>In this paper, we explore the generalized Schur decomposition in tensor formats utilizing the tensor-tensor product. We introduce the concept of the tensor generalized Schur decomposition and present its applications in various fields. Specifically, we apply this decomposition to optical cryptosystems, including asymmetric image cryptosystems and watermarking, as well as to multilinear discriminant analysis. Additionally, we investigate the normwise and componentwise perturbation bounds associated with this decomposition. Furthermore, we propose a tensor generalized Schur method to solve tensor-based generalized Sylvester equations, accompanied by an analysis of its backward error and perturbation bounds.</p>