This paper proposes a novel finger-individuating exoskeleton system with a non-contact leader-follower control strategy that effectively combines motion functionality and individual adaptability. Our solution comprises the following two interactive components: the leader side and the follower side. The leader side processes joint angle information from the healthy hand during motion via a Leap Motion Controller as the system input, providing more flexible and active operations owing to the non-contact manner. Then, as the follower side, the exoskeleton is driven to assist the user's hand for rehabilitation training according to the input. The exoskeleton mechanism is designed as a universal module that can adapt to various digit sizes and weighs only 40 g. Additionally, the current motion of the exoskeleton is fed back to the system in real time, forming a closed loop to ensure control accuracy. Finally, four experiments validate the design effectiveness and motion performance of the proposed exoskeleton system. The experimental results indicate that our prototype can provide an average force of about 16.5 N for the whole hand during flexing, and the success rate reaches 82.03% in grasping tasks. Importantly, the proposed prototype holds promise for improving rehabilitation outcomes, offering diverse options for different stroke stages or application scenarios.