The proliferation of inverter-based resources challenges traditional microgrid protection by introducing variable fault currents and complex transients. This paper presents a statistically adaptive differential protection scheme based on Kullback-Leibler divergence, implemented via a Bartlett-corrected G-statistic computed on logarithm-transformed current magnitudes. The method is a multivariate fault detection engine that employs the Mahalanobis distance to distinguish healthy and faulty states, enabling robust detection even in noisy environments. Detection thresholds are statistically derived from a chi-squared distribution for precise control over the false alarm rate. Upon detection, a lightweight classifier identifies the fault type by assessing per-phase G-statistics against dedicated thresholds, enhanced by a temporal persistence filter for security. Extensive simulations on a modified CIGRE 14-bus microgrid show high efficacy: sub-cycle average detection delays, high detection and classification accuracy across operating modes, resilience to high-impedance faults up to 250 Ohms, tolerance to 10 ms communication delay, and noise levels down to a 20 dB signal-to-noise ratio. These findings demonstrate a reproducible and computationally efficient solution for next-generation AC microgrid protection.