Deep learning has been widely applied for the channel state information (CSI) feedback in frequency division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems. The classical single-task training approach is implemented for the feedback model training in each channel scenario, and the requirements of large-amount task-specific data can hardly be satisfied. The huge training cost and storage usage of the model in multiple scenarios also hinder the application in practical systems. In this letter, a multi-task training approach is proposed to improve the feasibility of the feedback network. An encoder-shared feedback architecture and the training procedure are further designed to facilitate the implementation of the proposed approach. The experimental results indicate that the multi-task training approach can achieve comprehensive feedback performance with considerable reduction of training cost and storage usage of the feedback model. The source code for the experiments is available at GitHub.