Demographic attributes such as age, sex, and race can be predicted from medical images, raising concerns about bias in clinical AI systems. In brain MRI, this signal may arise from anatomical variation, acquisition-dependent contrast differences, or both, yet these sources remain entangled in conventional analyses. Without disentangling them, mitigation strategies risk failing to address the underlying causes. We propose a controlled framework based on disentangled representation learning, decomposing brain MRI into anatomy-focused representations that suppress acquisition influence and contrast embeddings that capture acquisition-dependent characteristics. Training predictive models for age, sex, and race on full images, anatomical representations, and contrast-only embeddings allows us to quantify the relative contributions of structure and acquisition to the demographic signal. Across three datasets and multiple MRI sequences, we find that demographic predictability is primarily rooted in anatomical variation: anatomy-focused representations largely preserve the performance of models trained on raw images. Contrast-only embeddings retain a weaker but systematic signal that is dataset-specific and does not generalise across sites. These findings suggest that effective mitigation must explicitly account for the distinct anatomical and acquisition-dependent origins of the demographic signal, ensuring that any bias reduction generalizes robustly across domains.