Federated learning (FL) is increasingly adopted for training finger vein recognition models to address client-level data scarcity while preserving privacy. However, as biometric data involves sensitive personal information, higher security guarantees are imperative for FL systems. This paper proposes a novel cloud-edge-end federated learning framework specifically designed for finger vein recognition, incorporating a Multi-dimensional Secret Sharing (MSS) scheme to secure model aggregation. MSS strategically integrates client additive secret sharing and edge Newton interpolation-based secret sharing, establishing a strict security threshold. Crucially, the framework guarantees secure aggregation under the non-collusion of at least a predefined number of edge servers, preventing compromise even if multiple clients or a single edge server are compromised. Evaluations demonstrate that the proposed framework not only provides robust security against collusion attacks but also improves finger vein model accuracy under challenging conditions of non-IID (non-independent and identically distributed) and class-imbalanced data. Furthermore, its effectiveness extends to multi-modal biometric systems (e.g., iris, face recognition), maintaining comparable accuracy to FedAvg while significantly enhancing aggregation security.