Multiplex heterogeneous networks are common in real-world scenarios, where entities interact through diverse types of relations across multiple semantic layers. Recent advances in multiplex heterogeneous graph neural networks have achieved remarkable results by incorporating node and relation types into message passing and designing relation-aware architectures. However, most existing methods either decouple relations and risk losing complex semantics or require handcrafted relation patterns, which limit scalability. Moreover, prevailing models are typically restricted to Euclidean space, making it difficult to capture non-Euclidean topologies and to distinguish complex interactions among heterogeneous nodes and relations. Standard GNN message passing, grounded in the homophily assumption, also proves inadequate for the intricate, coupled structures in multiplex heterogeneous graphs. To address these challenges, we propose MRiemGNN, a novel multiplex heterogeneous graph neural network that synergizes Euclidean and Riemannian spaces through a geometry-aware, relation-specific message passing scheme and cross-space mutual learning. Experiments on multiple real-world datasets show that MRiemGNN achieves superior performance, efficiency, and scalability on both node classification and link prediction tasks. The code of our model is provided in the appendix.