A key challenge in visible-infrared person re-identification (V-I ReID) is training a backbone model capable of effectively addressing the significant discrepancies across modalities. State-of-the-art methods that generate a single intermediate bridging domain are often less effective, as this generated domain may not adequately capture sufficient common discriminant information. This paper introduces the Bidirectional Multi-step Domain Generalization (BMDG), a novel approach for unifying feature representations across diverse modalities. BMDG creates multiple virtual intermediate domains by finding and aligning body part features extracted from both I and V modalities. Indeed, BMDG aims to reduce the modality gaps in two steps. First, it aligns modalities in feature space by learning shared and modality-invariant body part prototypes from V and I images. Then, it generalizes the feature representation by applying bidirectional multi-step learning, which progressively refines feature representations in each step and incorporates more prototypes from both modalities. In particular, our method minimizes the cross-modal gap by identifying and aligning shared prototypes that capture key discriminative features across modalities, then uses multiple bridging steps based on this information to enhance the feature representation. Experiments conducted on challenging V-I ReID datasets indicate that our BMDG approach outperforms state-of-the-art part-based models or methods that generate an intermediate domain from V-I person ReID.