In order to promote new energy consumption and low-carbon economic operation in microgrids, a multi-energy microgrid cluster model with shared energy storage power stations is constructed. Firstly, a multi-energy microgrid model with electricity, heat and gas is established, and multiple microgrids form a cooperative alliance to jointly establish shared energy storage. Secondly, acknowledging the intricate correlation and inherent unpredictability of wind and solar energy outputs, a refined multi-objective robust optimization scheduling model for the multi-energy microgrid cluster is formulated. This approach integrates Copula-multi-scenario confidence gap decision theory to attain the dual objectives of cost minimization and carbon emission reduction. Finally, considering the energy interaction contribution, risk level, and carbon reduction contribution of the members in the alliance, the benefit of the improved Shapely value method is used to quantify the contribution of each alliance subject and equitably distribute the cooperation surplus. The example results show that the proposed model reduces the total cost by 18.64% and carbon emission by 30.24% compared with the independent allocation of energy storage, and it realizes the precise matching between the comprehensive contribution of each subject to the alliance and the distribution of the cooperation surplus, so as to enhance the enthusiasm of the multi-energy microgrids to participate in the cooperation.