Do visual neural networks learn brain-aligned representations because they share architectural constraints and task objectives with biological vision or because they share universal features of natural image processing? We characterized the universality of hundreds of thousands of representational dimensions from networks with different architectures, tasks, and training data. We found that diverse networks learn to represent natural images using a shared set of latent dimensions, despite having highly distinct designs. Next, by comparing these networks with human brain representations measured with functional magnetic resonance imaging, we found that the most brain-aligned representations in neural networks are those that are universal and independent of a network’s specific characteristics. Each network can be reduced to fewer than 10 of its most universal dimensions with little impact on its representational similarity to the brain. These results suggest that the underlying similarities between artificial and biological vision are primarily governed by a core set of universal representations that are convergently learned by diverse systems.