Advanced autonomous driving systems require accurate vehicle dynamics modeling. However, identifying a precise dynamics model remains challenging due to strong nonlinearities and the coupled longitudinal and lateral dynamic characteristics. Previous research has employed physics-based analytical models or neural networks to construct vehicle dynamics representations. Nevertheless, these approaches often struggle to simultaneously achieve satisfactory performance in terms of system identification efficiency, modeling accuracy, and compatibility with linear control strategies. In this paper, we propose a fully data-driven dynamics modeling method tailored for complex distributed electric-drive trucks (DETs), leveraging Koopman operator theory to represent highly nonlinear dynamics in a lifted linear embedding space. To achieve high-precision modeling, we first propose a novel dual-branch encoder which encodes dynamic states and provides a powerful basis for the proposed Koopman-based methods entitled KODE. A physics-informed supervision mechanism, grounded in the geometric consistency of temporal vehicle motion, is incorporated into the training process to facilitate effective learning of both the encoder and the Koopman operator. Furthermore, to accommodate the diverse driving patterns of DETs, we extend the vanilla Koopman operator to a mixture-of-Koopman operator framework, enhancing modeling capability. Simulations conducted in a high-fidelity TruckSim environment and real-world experiments demonstrate that the proposed approach achieves state-of-the-art performance in long-term dynamics state estimation.