Existing research has shown that people often attribute human-like attributes to robots, which is generally known as the 'anthropomorphism' phenomenon. We use the notion of 'multi-dimensional anthropomorphism,' to perform a more fine-grained analysis of anthropomorphism in relation to robots in terms of several dimensions (e.g., uniquely and typically human, being alive or not, having emotions or not). Additionally, we expand on existing work, which has mostly focused on organism-based robot designs, by including object-based robot designs in our study of robot anthropomorphism. The results of an online survey study with 775 U.S. (393) and Chinese (382) participants show how people's personal characteristics (e.g., nationality) affect their perceptions of the anthropomorphism of robots, and how such perceptions differ between organism- and object-based robot designs. The effect on people's multi-dimensional anthropomorphism perceptions suggests new design implications for robotic technologies.