With the development of Internet of Medical Things, massive healthcare sensor data (HSD) are transmitted in the Internet, which faces various security problems. Healthcare data are sensitive and important for patients. Automatic classification of HSD has significant value for protecting the privacy of patients. Recently, the edge computing-based federated learning has brought new opportunities and challenges. It is difficult to develop a lightweight HSD classification system for edge computing. In particular, the classification system should consider the dynamic characteristics of HSD, e.g., the change of data distributions and the appearance of initially unknown classes. To solve these problems, the paper proposes a scalable and transferable classification system, called SCALT. It is a one-classifier-per-class system based on federated learning. It comprises a one-dimensional convolution-based network for feature extraction, and an individual mini-classifier for each class. It is easy to be scaled when new class appears since only a mini-classifier will be trained. The feature extractor is updated only when it is transferred to a new task. SCALT has a parameter protection mechanism, which can avoid catastrophic forgetting in sequential HSD classification tasks. We conduct comprehensive experiments to evaluate SCALT on three different physiological signal datasets: Electrocardiogram, Electroencephalogram and Photoplethysmograph. The accuracies on the three datasets are 98.65%, 91.10% and 89.93% respectively, which are higher than the compared state-of-the-art works. At last, an application of applying SCALT to protect the privacy of patients is presented.