Most state-of-the-art vein recognition methods rely on closed-set classification, which inherently limits their scalability and prevents the adaptive enrollment of new users without complete model retraining. We rigorously evaluate the computational boundaries of Deep Metric Learning (DML) under strict open-set constraints. Unlike standard closed-set approaches, we analyze the impact of data scarcity and domain shift on recognition performance. Our approach learns discriminative L2-normalised embeddings and employs prototype-based matching with a calibrated similarity threshold to effectively distinguish between enrolled users and unseen impostors. We evaluate the framework under a strict subject-disjoint protocol across four diverse datasets covering finger, wrist, and dorsal hand veins (MMCBNU 6000, UTFVP, FYO, and a dorsal hand-vein dataset). On the large-scale MMCBNU 6000 benchmark, our best model (ResNet50-CBAM) achieves an OSCR of 0.9945, AUROC of 0.9974, and EER of 1.57%, maintaining high identification accuracy (99.6% Rank-1) while robustly rejecting unknown subjects. Cross-dataset experiments evaluate the framework's generalisation across different acquisition setups, confirming that while the model handles large-scale data robustly, performance remains sensitive to domain shifts in low-data regimes. Ablation studies demonstrate that triplet-based objectives combined with a simple 1-NN classifier offer an optimal trade-off between accuracy and efficiency, enabling real-time deployment on commodity hardware.