Anthropomorphic underactuated hands are valued for their structural simplicity and inherent adaptability. However, the uncertainty arising from interdependent joint motions makes it challenging to capture various grasp states during hand-object interaction without increasing structural complexity through multiple embedded sensors. This motivates the need for an approach that can extract rich grasp-state information from a single sensing source while preserving the simplicity of underactuation. This study proposes an anthropomorphic underactuated hand that achieves comprehensive grasp state estimation, using only tendon-based proprioception provided by series elastic actuators (SEAs). Our approach is enabled by the design of a compact SEA with high accuracy and reliability that can be seamlessly integrated into sensorless fingers. By coupling accurate proprioceptive measurements with potential energy-based modeling, the system estimates multiple key grasp state variables, including contact timing, joint angles, relative object stiffness, and external disturbances. Finger-level experimental validations and extensive hand-level grasp functionality demonstrations confirmed the effectiveness of the proposed approach. These results highlight tendon-based proprioception as a compact and robust sensing modality for practical manipulation without reliance on vision or tactile feedback.