As large language models become increasingly capable at various writing tasks, their weakness at generating unique and creative content becomes a major liability. Although LLMs have the ability to generate text covering diverse topics, there is an overall sense of repetitiveness across texts that we aim to formalize and quantify via a similarity metric. The familiarity between documents arises from the persistence of underlying discourse structures. However, existing similarity metrics dependent on lexical overlap and syntactic patterns largely capture $\textit{content}$ overlap, thus making them unsuitable for detecting $\textit{structural}$ similarities. We introduce an abstraction based on linguistic theories in Questions Under Discussion (QUD) and question semantics to help quantify differences in discourse progression. We then use this framework to build $\textbf{QUDsim}$, a similarity metric that can detect discursive parallels between documents. Using QUDsim, we find that LLMs often reuse discourse structures (more so than humans) across samples, even when content differs. Furthermore, LLMs are not only repetitive and structurally uniform, but are also divergent from human authors in the types of structures they use.