The development of cloud computing has facilitated the transition of biometric authentication technology from traditional standalone applications to network-based services, while also raising concerns about privacy breaches in the transmission and storage of biometric data. To response, we propose a secure cloud-edge collaborative biometric authentication framework via two-factor secret sharing, named Triplet-Bio. Inspired by the characteristics(distributed, real-time, security, etc.) of edge computing paradigm, Triplet-Bio encrypts biometric data at the edge for transmission to the cloud, thereby reducing transmission costs and enhancing security. Furthermore, we design a two-factor secret sharing scheme integrating biometric data and hardware fingerprints(Physical Unclonable Functions, PUFs). Successful identification and service authorization are contingent upon the correctness of both factors and their mutual validation among the three parties involved. Finally, a comprehensive evaluation of the proposed scheme is conducted from the perspectives of biometric recognition performance and security analysis, thereby demonstrating the efficiency and security of Triplet-Bio.