Users’ increasing adoption of voice assistant services is fostering the growth of a novel strand of marketing research on the branding implications of brand anthropomorphism (BA). However, the branding outcomes of brand anthropomorphization in this research area remain underinvestigated. Accordingly, in the name-brand voice assistant (NBVA) interaction field, this study tests a model of the consequences of brand anthropomorphism, outlining the relationships among brand anthropomorphism, brand trust, and multidimensional consumer–brand engagement (CBE), i.e., the relevant cognitive, affective, and behavioral dimensions, as well as the moderating role of perceived privacy risk. A survey of young adults shows that brand anthropomorphism positively affects brand trust as well as the affective and behavioral dimensions of CBE. Furthermore, perceived privacy risk positively moderates the relationship between brand anthropomorphism and brand trust. Specifically, the influence of brand anthropomorphism on brand trust is strengthened at higher levels of perceived privacy risk. This article thus enriches the understanding of brand anthropomorphism and user VA response by exploring underresearched branding outcomes of BA in the context of NBVA interaction.