Accurately understanding the Tire-Road Friction Coefficient (TRFC) interaction is crucial for enhancing overall vehicle control performance. However, existing estimation methods heavily rely on the accuracy of tire modeling, which may not predict TRFC effectively under poor modeling accuracy or changing operating conditions. This paper proposes an interacting multiple-model TRFC estimation method using tire force observation. Firstly, taking advantage of the observation capabilities of distributed drive electric vehicles, a longitudinal force estimator with unknown inputs is designed. Simultaneously, a lateral force observer based on adaptive sliding mode observer (ASMO) is developed, fully utilizing the onboard sensor data of the vehicle. A TRFC estimator with square root cubature Kalman filter, incorporating square root filtering, is proposed to reduce algorithm complexity while ensuring accuracy. Finally, an interacting multiple-model mechanism is specifically developed for both pure longitudinal dynamics and combined conditions. Through Carsim-Matlab co-simulation and high TRFC test conditions of a real vehicle equipped with a 6-axis wheel transducer, it is shown that the algorithm proposed in this article can accurately estimate tire force and TRFC under various maneuvering conditions and different TRFC conditions. Based on the comparison with traditional algorithms, it shows that our proposed algorithm has higher estimation accuracy and robustness.