Ensuring safe, robust, and scalable motion planning for multi-agent systems in dynamic and uncertain environments is a persistent challenge, driven by complex inter-agent interactions, stochastic disturbances, and model uncertainties. To overcome these challenges, particularly the computational complexity of coupled decision-making and the need for proactive safety guarantees, we propose a Reachability-Enhanced Dynamic Potential Game (RE-DPG) framework, which integrates game-theoretic coordination into reachability analysis. This approach formulates multi-agent coordination as a dynamic potential game, where the Nash equilibrium (NE) defines optimal control strategies across agents. To enable scalability and decentralized execution, we develop a Neighborhood-Dominated iterative Best Response (ND-iBR) scheme, built upon an iterated $\varepsilon$-BR (i$\varepsilon$-BR) process that guarantees finite-step convergence to an $\varepsilon$-NE. This allows agents to compute strategies based on local interactions while ensuring theoretical convergence guarantees. Furthermore, to ensure safety under uncertainty, we integrate a Multi-Agent Forward Reachable Set (MA-FRS) mechanism into the cost function, explicitly modeling uncertainty propagation and enforcing collision avoidance constraints. Through both simulations and real-world experiments in 2D and 3D environments, we validate the effectiveness of RE-DPG across diverse operational scenarios.