Shared energy storage, as an emerging energy storage solution, helps to integrate renewable energy sources within microgrids and reduce operational costs, unleashing the potential for resource sharing among independent stakeholders in the microgrid. However, traditional approaches to shared energy storage and interconnection between microgrids overlook the issue of information privacy in transactions among entities, and cooperative strategies often fail to achieve fair benefit allocation. To address this, a distributed robust game-theoretic optimization scheduling method is proposed for microgrid clusters with shared energy storage. First, a microgrid model with multiple energy forms and a shared energy storage model are established. Second, to mitigate the impact of uncertain wind and solar power outputs on system economics, robust optimization theory is applied to handle uncertainty and solve for the worst-case probability distribution of operational strategies. Finally, based on the Nash bargaining theory, a joint operation model for shared energy storage and microgrid systems is developed, and the model is decomposed into two sub-problems: minimizing the joint system operational cost and negotiating internal electricity transactions within the system, using the alternating direction method of multipliers with good convergence and privacy properties. Through comparative analysis before and after cooperation, the proposed method reduces microgrid operational costs by 2.99%, 4.90%, and 4.27%, respectively, demonstrating its effectiveness in addressing wind and solar power uncertainty while reducing operational costs for all stakeholders, achieving both flexibility and economic efficiency in the system.