Deep stereo matching has advanced significantly on benchmark datasets through fine-tuning but falls short of the zero-shot generalization seen in foundation models in other vision tasks. We introduce CogStereo, a novel framework that addresses challenging regions, such as occlusions or weak textures, without relying on dataset-specific priors. CogStereo embeds implicit spatial cognition into the refinement process by using monocular depth features as priors, capturing holistic scene understanding beyond local correspondences. This approach ensures structurally coherent disparity estimation, even in areas where geometry alone is inadequate. CogStereo employs a dual-conditional refinement mechanism that combines pixel-wise uncertainty with cognition-guided features for consistent global correction of mismatches. Extensive experiments on Scene Flow, KITTI, Middlebury, ETH3D, EuRoc, and real-world demonstrate that CogStereo not only achieves state-of-the-art results but also excels in cross-domain generalization, shifting stereo vision towards a cognition-driven approach.