Autonomous mobility is emerging as a new disruptive mode of urban transportation for moving cargo and passengers. However, designing scalable autonomous fleet coordination schemes to accommodate fast-growing mobility systems is challenging primarily due to the increasing heterogeneity of the fleets, time-varying demand patterns, service area expansions, and communication limitations. We introduce the concept of partially observable advanced air mobility games to coordinate a fleet of aerial vehicles by accounting for the heterogeneity of the interacting agents and the self-interested nature inherent to commercial mobility fleets. To model the complex interactions among the agents and the observation uncertainty in the mobility networks, we propose a novel heterogeneous graph attention encoder-decoder (HetGAT Enc-Dec) neural network-based stochastic policy. We train the policy by leveraging deep multi-agent reinforcement learning, allowing decentralized decision-making for the agents using their local observations. Through extensive experimentation, we show that the learned policy generalizes to various fleet compositions, demand patterns, and observation topologies. Further, fleets operating under the HetGAT Enc-Dec policy outperform other state-of-the-art graph neural network policies by achieving the highest fleet reward and fulfillment ratios in on-demand mobility networks.