Shared energy storage system provides an attractive solution to the high configuration cost and low utilization rate of multi-microgrid energy storage system. In this paper, an electricity-heat integrated energy storage supplier (EHIESS) containing electricity and heat storage devices is proposed to provide shared energy storage services for multi-microgrid system in order to realize mutual profits for different subjects. To this end, electric boiler (EB) is introduced into EHIESS to realize the electricity-heat coupling of EHIESS and improve the energy utilization rate of electricity and heat storage equipment. Secondly, due to the problem of the uncertainty in user-side operation of multi-microgrid system, a price-based demand response (DR) mechanism is proposed to further optimize the resource allocation of shared electricity and heat energy storage devices. On this basis, a bi-level optimization model considering the capacity configuration of EHIESS and the optimal scheduling of multi-microgrid system is proposed, with the objectives of maximizing the profits of energy storage suppliers in upper-level and minimizing the operation costs of the multi-microgrid system in lower-level, and solved based on the Karush-Kuhn-Tucker (KKT) condition and Big-M method. The simulation results show that in case of demand response, the total operation cost of multi-microgrid system and the total operation profit of EHIESS are 51,687.73 and 11,983.88 CNY, respectively; and the corresponding electricity storage unit capacity is 9730.80 kWh. The proposed model realizes the mutual profits of EHIESS and multi-microgrid system.