Robot skill acquisition processes driven by reinforcement learning often rely on simulations to efficiently generate large-scale interaction data. However, the absence of simulation models for tactile sensors has hindered the use of tactile sensing in such skill learning processes, limiting the development of effective policies driven by tactile perception. To bridge this gap, we present TwinTac, a system that combines the design of a physical tactile sensor with its digital twin model. Our hardware sensor is designed for high sensitivity and a wide measurement range, enabling high quality sensing data essential for object interaction tasks. Building upon the hardware sensor, we develop the digital twin model using a real-to-sim approach. This involves collecting synchronized cross-domain data, including finite element method results and the physical sensor's outputs, and then training neural networks to map simulated data to real sensor responses. Through experimental evaluation, we characterized the sensitivity of the physical sensor and demonstrated the consistency of the digital twin in replicating the physical sensor's output. Furthermore, by conducting an object classification task, we showed that simulation data generated by our digital twin sensor can effectively augment real-world data, leading to improved accuracy. These results highlight TwinTac's potential to bridge the gap in cross-domain learning tasks.