Endovascular procedures have revolutionized vascular disease treatment, yet their manual execution is challenged by the demands for high precision, operator fatigue, and radiation exposure. Robotic systems have emerged as transformative solutions to mitigate these inherent limitations. A pivotal moment has arrived, where a confluence of pressing clinical needs and breakthroughs in AI creates an opportunity for a paradigm shift toward Embodied Intelligence (EI), enabling robots to navigate complex vascular networks and adapt to dynamic physiological conditions. Data-driven approaches, leveraging advanced computer vision, medical image analysis, and machine learning, drive this evolution by enabling real-time vessel segmentation, device tracking, and anatomical landmark detection. Reinforcement learning and imitation learning further enhance navigation strategies and replicate expert techniques. This review systematically analyzes the integration of EI into endovascular robotics, identifying profound systemic challenges such as the heterogeneity in validation standards and the gap between human mimicry and machine-native capabilities. Based on this analysis, a conceptual roadmap is proposed that reframes the ultimate objective away from systems that supplant clinical decision-making. This vision of augmented intelligence, where the clinician's role evolves into that of a high-level supervisor, provides a principled foundation for the future of the field.