Unsupervised visible-infrared person re-identification (USL-VI-ReID) is a promising yet highly challenging retrieval task. The key challenges in USL-VI-ReID are to accurately generate pseudo-labels and establish pseudo-label correspondences across modalities without relying on any prior annotations. Recently, clustered pseudo-label methods have gained more attention in USL-VI-ReID. However, most existing methods don't fully exploit the intra-class nuances, as they simply utilize a single memory that represents an identity to establish cross-modality correspondences, resulting in noisy cross-modality correspondences. To address the problem, we propose a Multi-Memory Matching (MMM) framework for USL-VI-ReID. We first design a simple yet effective Cross-Modality Clustering (CMC) module to generate the pseudo-labels through clustering together both two modality samples. To associate cross-modality clustered pseudo-labels, we design a Multi-Memory Learning and Matching (MMLM) module, ensuring that optimization explicitly focuses on the nuances of individual perspectives and establishes reliable cross-modality correspondences. Finally, we design a Soft Cluster-level Alignment (SCA) loss to narrow the modality gap while mitigating the effect of noisy pseudo-labels through a soft many-to-many alignment strategy. Extensive experiments on the public SYSU-MM01 and RegDB datasets demonstrate the reliability of the established cross-modality correspondences and the effectiveness of MMM. The source codes will be released.