Soft robotic systems pose a significant challenge for traditional modeling, estimation, and control approaches, primarily owing to their inherent complexity and virtually infinite degrees of freedom (DoFs). This work introduces an innovative method for dynamically estimating the states of tendon-actuated soft manipulators. Our technique merges the Geometric Variable-Strain (GVS) approach with a kinematic formula that links the length variation of tendons to the deformations of the manipulator and a nonlinear observer design based on state-dependent Riccati equation (SDRE). In our methodology, the soft links are represented by Cosserat rods, and the robot's geometry is parameterized by the strain field along its length. Consequently, its infinite dimensions can be described by utilizing multiple degrees of freedom, depending on the required precision. This enables us to estimate the states (pose and velocity) of tendon-actuated soft manipulators solely based on tendon displacements and actuator forces. Through simulation, we demonstrate the convergence of our estimation method across various DoFs and actuator numbers, revealing a trade-off between the number of DoFs and required actuators for observing system states. Furthermore, we validate our approach with an experimental prototype of 25 cm in length, achieving an average tip position error during dynamic motion of 1.79 cm-less than 7% of the overall body length.