The article analyses critical blindspots in current European Artificial Intelligence (AI) policies and examines the potential impact of data and AI in the emerging socio-technical ecosystem of the contemporary Media and Communications (MC) sector from the perspective of critical media and communication studies. We first identify central blind spots in the dominant EU trustworthy and risk-based approach to governing AI. Next, we propose a novel multi-level framework to analyse key policy challenges for governing AI in the MC sector. The framework and discussion are based on desk research and multi-stakeholder expert discussions. The article concludes with reflections on AI governance in development, deployment and use in the MC sector.