Fluctuation statistics of quantum many-body systems are very revealing. In driven-dissipative systems exhibiting macroscopic quantum coherence, exciton polaritons condense under incoherent pumping, so that the phase dynamics can be mapped to the stochastic Kardar-Parisi-Zhang (KPZ) equation. However, in two dimensions (2D), it has been theoretically argued that the KPZ regime may be disturbed by the presence of eddies, and non-equilibrium BKT behavior has been reported near the condensation threshold. Here we show that the universal KPZ property emerges when considering a discretized 2D polariton system. We support our analysis by extensive numerical simulations of the discrete stochastic generalized Gross-Pitaevsky equation. We show that the first-order correlation function of the condensate exhibits a stretched exponential behavior in space and time, with a critical exponent characteristic of the 2D KPZ universality class, suggesting that the associated scaling function is the theoretical It indicates an exact match for a function that is We also obtained the distribution of the phase fluctuations and confirmed that they are non-Gaussian, as expected for a KPZ stochastic process.