Temporal graphs serve as a critical foundation for modeling evolving interactions in domains ranging from financial networks to social media. Mining temporal motifs is essential for applications such as fraud detection, cybersecurity, and dynamic network analysis. However, conventional motif mining approaches treat each query independently, incurring significant redundant computations when similar substructures exist across multiple motifs. In this paper, we propose Mayura, a novel framework that unifies the mining of multiple temporal motifs by exploiting their inherent structural and temporal commonalities. Central to our approach is the Motif-Group Tree (MG-Tree), a hierarchical data structure that organizes related motifs and enables the reuse of common search paths, thereby reducing redundant computation. We propose a co-mining algorithm that leverages the MG-Tree and develop a flexible runtime capable of exploiting both CPU and GPU architectures for scalable performance. Empirical evaluations on diverse real-world datasets demonstrate that Mayura achieves substantial improvements over the state-of-the-art techniques that mine each motif individually, with an average speed-up of 2.4x on the CPU and 1.7x on the GPU, while maintaining the exactness required for high-stakes applications.