While service robots are transforming industries, existing studies often assume a high degree of immunity to the influence of situated physical environments. This assumption may not hold in the context of robotic services, in which physical environments (where a service robot operates, hereafter, physical service context) contribute significantly to service design. This study employs the Privacy Calculus Theory to develop a theoretical framework for interpreting the impacts of physical service context on robotic service adoption. It investigates the influence of anthropomorphism on usage intention through the dual pathway of perceived privacy invasion and trust across three distinct contexts: home, workspace, and public spaces. By a large-scale scenario-based experiment involving 3893 participants, we identified three non-linear relationships: a fluctuating trajectory between anthropomorphism and perceived privacy invasion, a non-linear positive monotonic relationship between anthropomorphism and trust, and an uncanny valley (UV)-like pattern between anthropomorphism and usage intention. The results reveal that usage intention varies significantly across physical service contexts, being lowest at home. The influences of both trust and perceived privacy invasion on usage intention are highly context-dependent. These findings reconcile conflicting results in the literature by demonstrating that the effect of anthropomorphism is non-linear and context-bound. The study contributes by highlighting the critical role of physical service context and privacy perceptions for embodied artificial intelligence (AI) and offers clear guidance for designing and deploying service robots that balance human-like features with privacy-sensitive adoption strategies.