We introduce Distribution–Metric Geometry (DMG), a geometric framework for analyzing phase transitions directly in the space of empirical probability distributions generated by microscopic models. Instead of focusing on model-specific order parameters, DMG constructs a multi-metric embedding of each model into an information-geometric manifold. Phase transitions then appear as geometric events: sudden reorientation of the trajectory in metric space, peaks in geometric speed and curvature, and temporary expansion of intrinsic dimensionality. Across three classical 2D lattice models (Villain, XY, Ising), DMG reveals that their trajectories in metric space require, respectively, one, two, and three principal geometric modes. These intrinsic dimensions are stable under the choice of metrics and system size, and behave as robust geometric invariants of each model. . The framework is modelagnostic and extends naturally to complex systems where traditional order parameters are unknown or purely topological.