Federated learning is a learning paradigm to enable collaborative learning across different parties without revealing raw data. Notably, vertical federated learning (VFL), where parties share the same set of samples but only hold partial features, has a wide range of real-world applications. However, most existing studies in VFL disregard the "record linkage'' process. They design algorithms either assuming the data from different parties can be exactly linked or simply linking each record with its most similar neighboring record. These approaches may fail to capture the key features from other less similar records. Moreover, such improper linkage cannot be corrected by training since existing approaches provide no feedback on linkage during training. In this paper, we design a novel coupled training paradigm, FedSim, that integrates one-to-many linkage into the training process. Besides enabling VFL in many real-world applications with fuzzy identifiers, FedSim also achieves better performance in traditional VFL tasks. Moreover, we theoretically analyze the additional privacy risk incurred by sharing similarities. Our experiments on eight datasets with various similarity metrics show that FedSim outperforms other state-of-the-art baselines. The codes of FedSim are available at https://github.com/Xtra-Computing/FedSim.