A synchronous generator's P-Q envelope is shaped by thermal limits in the stator, the rotor, and the end-region core. A battery storage system's envelope is shaped by something else entirely: inverter electronics, battery state-of-charge, and DC-bus voltage.
The biggest single difference is that a battery operates in four quadrants of the P-Q plane rather than two. A synchronous generator generally exports real power and exchanges reactive either direction — two quadrants. A battery actively cycles between charge and discharge. All four quadrants are operating points the regulator expects to be tested.