Autonomous mobility is emerging as a new mode of urban transportation for moving cargo and passengers. However, such fleet coordination schemes face significant challenges in scaling to accommodate fast-growing fleet sizes that vary in their operational range, capacity, and communication capabilities. We introduce the concept of partially observable advanced air mobility games to coordinate a fleet of aerial vehicle agents accounting for their heterogeneity and self-interest inherent to commercial mobility fleets. We propose a novel heterogeneous graph attention-based encoder-decoder (HetGAT Enc-Dec) neural network to construct a generalizable stochastic policy stemming from the inter- and intra-agent relations within the mobility system. We train our 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 fleets operating under the HetGAT Enc-Dec policy outperform other state-of-the-art graph neural network-based policies by achieving the highest fleet reward and fulfillment ratios in an on-demand mobility network.