The lack of battery state and degradation awareness in electric vehicles (EVs) participating in vehicle-to-grid (V2G) frequency response (FR) can accelerate battery degradation and impair frequency support. To address this issue, we propose a distributed model predictive control (DMPC) strategy based on battery state and degradation awareness for V2G FR in charging stations (CSs), explicitly integrating battery state and degradation into the proposed strategy. First, a resistor-capacitor (RC) equivalent circuit model of batteries is integrated into the V2G FR. This allows the controller to capture internal battery states including state-of-charge and battery degradation rate. The model also monitors signals such as voltage and current, thereby enhancing both model fidelity and practical implementability. Second, an equivalent degradation model of batteries is embedded within the V2G FR, enabling the controller to jointly account for battery state and degradation while supporting V2G FR. Finally, a cross-scale hierarchical framework for V2G control strategy is developed to decouple and coordinate battery-level states and grid-level power dispatch, achieving hierarchical coordination across individual batteries and the CS. Simulation results demonstrate that the proposed DMPC strategy improves frequency regulation effectiveness, limits V2G FR tracking deviation, enhances battery operation safety and degradation preservation, and achieves more balanced V2G FR power allocation among EVs.