In this paper, we propose a digital twin (DT)-based user-centric approach for processing sensing data in an integrated sensing and communication (ISAC) system with high accuracy and efficient resource utilization. The considered scenario involves an ISAC device with a lightweight deep neural network (DNN) and a mobile edge computing (MEC) server with a large DNN. After collecting sensing data, the ISAC device either processes the data locally or uploads them to the server for higher-accuracy data processing. To cope with data drifts, the server updates the lightweight DNN when necessary, referred to as continual learning. Our objective is to minimize the long-term average computation cost of the MEC server by optimizing two decisions, i.e., sensing data offloading and sensing data selection for the DNN update. A DT of the ISAC device is constructed to predict the impact of potential decisions on the long-term computation cost of the server, based on which the decisions are made with closed-form formulas. Experiments on executing DNN-based human motion recognition tasks are conducted to demonstrate the outstanding performance of the proposed DT-based approach in computation cost minimization.