This paper presents a decentralized control framework that incorporates social awareness into multi-agent systems with unknown dynamics to achieve prescribed-time reach-avoid-stay tasks in dynamic environments. Each agent is assigned a social awareness index that quantifies its level of cooperation or self-interest, allowing heterogeneous social behaviors within the system. Building on the spatiotemporal tube (STT) framework, we propose a real-time STT framework that synthesizes tubes online for each agent while capturing its social interactions with others. A closed-form, approximation-free control law is derived to ensure that each agent remains within its evolving STT, thereby avoiding dynamic obstacles while also preventing inter-agent collisions in a socially aware manner, and reaching the target within a prescribed time. The proposed approach provides formal guarantees on safety and timing, and is computationally lightweight, model-free, and robust to unknown disturbances. The effectiveness and scalability of the framework are validated through simulation and hardware experiments on a 2D omnidirectional