How has the public responded to the increasing prevalence of artificial intelligence (AI)-based technologies? We investigate public perceptions of AI by collecting over 12,000 responses over 12 months from a nationally representative U.S. sample. Participants provided open-ended metaphors reflecting their mental models of AI, a methodology that overcomes the limitations of traditional self-reported measures by capturing more nuance. Using a mixed-methods approach combining quantitative clustering and qualitative coding, we identify 20 dominant metaphors shaping public understanding of AI. To analyze these metaphors systematically, we present a scalable framework integrating language modeling (LM)-based techniques to measure key dimensions of public perception: anthropomorphism (attribution of human-like qualities), warmth, and competence. We find that Americans generally view AI as warm and competent, and that over the past year, perceptions of AI's human-likeness and warmth have significantly increased ($+34\%, r = 0.80, p < 0.01; +41\%, r = 0.62, p < 0.05$). These implicit perceptions, along with the identified dominant metaphors, strongly predict trust in and willingness to adopt AI ($r^2 = 0.21, 0.18, p < 0.001$). Moreover, we uncover systematic demographic differences in metaphors and implicit perceptions, such as the higher propensity of women, older individuals, and people of color to anthropomorphize AI, which shed light on demographic disparities in trust and adoption. In addition to our dataset and framework for tracking evolving public attitudes, we provide actionable insights on using metaphors for inclusive and responsible AI development.