Tactile object recognition is vital for robotic handling systems; however, existing technologies that concentrate on tactile sensors with high modulus are not suitable for soft grippers to classify deformable objects. In this letter, we integrated an indenter layer into the traditional microfluidic tactile sensor to increase its sensitivity based on the lensing effect of human skin. For the application of the developed sensor, we built HustGripper, a tendon-driven soft gripper, where the sensor was bonded on the fingertip. Experiments on the tactile classification of the deformable objects were conducted to validate the performance of the sensor, where different indenters, exploratory procedures, and data processing approaches were considered to explore the key factor to determine the classification accuracy.