Efficient and low-carbon energy utilization is a crucial aspect of promoting green and sustainable development. Multi-energy microgrids, which incorporate multiple interchangeable energy types, offer effective solutions for low-carbon and efficient energy consumption. This study aims to investigate the sharing of energy and carbon in multiple multi-energy microgrids (MMEMs) to enhance their economic impact, low-carbon attributes, and the efficient utilization of renewable energy. In this paper, an energy–carbon co-sharing operation model is established, incorporating carbon capture systems (CCSs) and two-stage power-to-gas (P2G) devices within the MMEMs to actualize low-carbon operation. Furthermore, based on cooperative game theory, this paper establishes an energy–carbon co-sharing Nash negotiation model and negotiates based on the energy–carbon contribution of each subject in the cooperation as bargaining power so as to maximize both the benefits of the MMEM alliance and the distribution of the cooperation benefits. The case study results show that the overall benefits of the alliance can be increased through Nash negotiation. Energy–carbon co-sharing can effectively increase the renewable energy consumption rate of 8.34%, 8.78%, and 8.83% for each multi-energy microgrid, and the overall carbon emission reduction rate reaches 17.81%. Meanwhile, the distribution of the benefits according to the energy–carbon co-sharing contribution capacity of each entity is fairer.