Finding latent partitions in complex systems is essential for a more accurate and comprehensive understanding of their relationships and structures. This is particularly true for those that exhibit diverse types of interactions, such as multiplex networks, where researchers can uncover hidden patterns and communities that may be overlooked in traditional single-layer network analyses. In this work, we propose a novel Bayesian nonparametric variation of the stochastic block model tailored to multiplex networks. The dependency across different layers is modelled by encoding all the possible combinations of relationships among the nodes in suitable categorical edge weights. The proposed technique is then applied to a multiplex network representing different relationships among powerful florentine families during the Renaissance.