Robots are often required to localize in environments with unknown object classes and semantic ambiguity. However, when performing global localization using semantic objects, high semantic ambiguity intensifies object misclassification and increases the likelihood of incorrect associations, which in turn can cause significant errors in the estimated pose. Thus, in this letter, we propose a multi-label likelihood-based semantic graph matching framework for object-level global localization. The key idea is to exploit multi-label graph representations, rather than single-label alternatives, to capture and leverage the inherent semantic context of object observations. Based on these representations, our approach enhances semantic correspondence across graphs by combining the likelihood of each node with the maximum likelihood of its neighbors via context-aware likelihood propagation. For rigorous validation, data association and pose estimation performance are evaluated under both closed-set and open-set detection configurations. In addition, we demonstrate the scalability of our approach to large-vocabulary object categories in both real-world indoor scenes and synthetic environments.