The rapid integration of Artificial Intelligence (AI) into organizational technology frameworks has transformed how organizations engage with AI-driven models, influencing both operational performance and strategic innovation. With the advent of foundation models, the importance of structured platforms for AI model exchange has become paramount for organizational efficacy and adaptability. However, a comprehensive framework to categorize and understand these platforms remains underexplored. To address this gap, our taxonomy provides a structured approach to categorize AI exchange platforms, examining key dimensions and characteristics, as well as revealing interesting interaction patterns between public research institutions and organizations: Some platforms leverage peer review as a mechanism for quality control, and provide mechanisms for online testing, deploying, and customization of models. Our paper is beneficial to practitioners seeking to understand challenges and opportunities that arise from AI exchange platforms. For academics, the taxonomy serves as a foundation for further research into the evolution, impact, and best practices associated with AI model sharing and utilization in different contexts. Additionally, our study provides insights into the evolving role of AI in various industries, highlighting the importance of adaptability and innovation in platform design. This paper serves as a critical resource for understanding the dynamic interplay between technology, business models, and user engagement in the rapidly growing domain of AI model exchanges pointing also towards possible future evolution.