The global population is inevitably aging due to increased life expectancy and declining birth rates, leading to an amplified demand for innovative social and healthcare services. One promising avenue is the introduction of companion robots. These robots are designed to provide physical assistance as well as emotional support and companionship, necessitating effective human–robot interaction (HRI). This study explores the role of cognitive empathy within HRI, focusing on the influence of robot facial animacy and emotional expressions on perspective-taking abilities—a key aspect of cognitive empathy—across different age groups. To this end, a director task involving 60 participants (30 young and 30 older adults) with varying degrees of robot facial animacy (0%, 50%, 100%) and emotional expressions (happy, neutral) was conducted. The results revealed that older adults displayed enhanced perspective-taking with higher animacy faces. Interestingly, while happiness on high-animacy faces improved perspective-taking, the same expression on low-animacy faces reduced it. These findings highlight the importance of considering facial animacy and emotional expressions in designing companion robots for older adults to optimize user engagement and acceptance. The study’s implications are pertinent to the design and development of socially effective service robots, particularly for the aging population.