Collaborative robotics combines human and robot capabilities in shared workspaces, integrating robotics, human-computer interaction, ergonomics, and psychology. In industrial settings, human–robot collaboration typically involves physical interaction with collaborative robots (cobots), designed to assist in tasks such as manufacturing and logistics. However, we discuss that cognitive and social-affective layers of interaction are equally crucial for enhancing collaboration. By adopting the 4E-cognition framework, we argue that the concept of social embodiment, widely used in virtual agents and service robotics, could contribute to the design and implementation of industrial cobots. While socially relevant functions are often secondary to operational efficiency, integrating them into cobot design could enhance interaction quality, trust, and overall task performance. Focusing on cobots’ social presence and their ability to engage in proactive interactions can improve their acceptance and facilitate a sense of shared intentionality and agency in human operators. Finally, we discuss the ethical and practical challenges of developing socially embodied humanoid cobots, addressing the balance between cobot autonomy and human control, adaptive learning in long-term interaction and personalization, and the implications of building trust and transparency in human–robot collaboration. A more holistic design approach could yield both operational benefits and improved human well-being in collaborative environments.