Stereo matching algorithms that leverage end-to-end convolutional neural networks have recently demonstrated notable advancements in performance. However, a common issue is their susceptibility to domain shifts, hindering their ability in generalizing to diverse, unseen realistic domains. We argue that existing stereo matching networks overlook the importance of extracting semantically and structurally meaningful features. To address this gap, we propose an effective hierarchical object-aware dual-level contrastive learning (HODC) framework for domain generalized stereo matching. Our framework guides the model in extracting features that support semantically and structurally driven matching by segmenting objects at different scales and enhances correspondence between intra- and inter-scale regions from the left feature map to the right using dual-level contrastive loss. HODC can be integrated with existing stereo matching models in the training stage, requiring no modifications to the architecture. Remarkably, using only synthetic datasets for training, HODC achieves state-of-the-art generalization performance with various existing stereo matching network architectures, across multiple realistic datasets.