Visible–infrared person re-identification (VI-ReID) aims to match pedestrian images across heterogeneous modalities. As a key technology in intelligent transportation systems, VI-ReID supports cross-camera tracking, behavior analysis, and security monitoring, particularly in nighttime or low-illumination scenarios. Despite recent advances, existing methods still encounter two critical challenges: (i) semantic misalignment between low-level and high-level features across modalities, and (ii) distribution discrepancies between visible and infrared images. To address these challenges, we propose a novel framework, Mutual Distillation Driven Dual-Space Matching (MDDM), which performs modality alignment in two complementary spaces. For challenge (i), we design a Dual Level Fusion (DLF) module to capture and adaptively fuse hierarchical features, aligning modalities by integrating both low- and high-level semantics across spatial and channel dimensions. In addition, a Modality Invariant Augmentation (MIA) module is developed to extract fine-grained semantic cues and enhance identity discrimination, thereby reinforcing the correlation between visible and infrared modalities and facilitating the learning of robust shared representations. For challenge (ii), we introduce Dual-Space Matching (DSM), which aligns features in both Hilbert and Euclidean spaces. Furthermore, a mutual distillation strategy is incorporated to promote cross-space consistency and alleviate modality-specific discrepancies. Extensive experiments on widely used VI-ReID benchmarks demonstrate the superiority and flexibility of the proposed method, which consistently achieves competitive performance across multiple datasets. Our code is available at https://github.com/lfjiang-cn/MDDM.