As Large Language Models (LLMs) become more and more successful at a wide range of tasks, their (in)ability to generalize remains difficult to quantify and poorly understood beyond limited domains. In particular, LLMs are known to struggle generalizing multilingually, to languages  
 poorly attested in their training data. To understand why this may be, and what enables some models to perform better than others, we turn to a long history of work across the cognitive sciences, arguing that successful generalization derives from appropriate representations in similarity space. We look at how well LLMs' representations capture the hierarchical similarity among distinct languages. Strikingly, we show LLMs' representations largely recover the hierarchical structure of the Indo-European language family tree -- grouping languages that are members of the same subfamily closely together in representation space. Furthermore, we show that the degree to which models reflect the similarity structure of  languages correlates with their performance on XNLI, a multilingual natural language inference benchmark. This extends classic work on similarity-driven generalization at scale, showing how models that represent similar languages similarly generalize better from one language to another.