Entity matching (EM) is a fundamental task in data integration and analytics, essential for identifying records that refer to the same real-world entity across diverse sources. In practice, datasets often differ widely in structure, format, schema, and semantics, creating substantial challenges for EM. We refer to this setting as Heterogeneous EM (HEM). This survey offers a unified perspective on HEM by introducing a taxonomy, grounded in prior work, that distinguishes two primary categories -- representation and semantic heterogeneity -- and their subtypes. The taxonomy provides a systematic lens for understanding how variations in data form and meaning shape the complexity of matching tasks. We then connect this framework to the FAIR principles -- Findability, Accessibility, Interoperability, and Reusability -- demonstrating how they both reveal the challenges of HEM and suggest strategies for mitigating them. Building on this foundation, we critically review recent EM methods, examining their ability to address different heterogeneity types, and conduct targeted experiments on state-of-the-art models to evaluate their robustness and adaptability under semantic heterogeneity. Our analysis uncovers persistent limitations in current approaches and points to promising directions for future research, including multimodal matching, human-in-the-loop workflows, deeper integration with large language models and knowledge graphs, and fairness-aware evaluation in heterogeneous settings.