Many state-of-the-art generative AI (GenAI) systems are increasingly prone to anthropomorphic behaviors, i.e., to generating outputs that are perceived to be human-like. While this has led to scholars increasingly raising concerns about possible negative impacts such anthropomorphic AI systems can give rise to, anthropomorphism in AI development, deployment, and use remains vastly overlooked, understudied, and underspecified. In this blog post, we argue that we cannot thoroughly map the social impacts of generative AI without mapping the social impacts of anthropomorphic AI, and outline a call to action.