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Hyperbolic uniform disk graphs (HUDGs) are intersection graphs of disks with some radius r in the hyperbolic plane, where r may be constant or depend on the number of vertices in a family of HUDGs. We show that HUDGs with constant clique number do not admit product structure, i.e., that there is no constant c such that every such graph is a subgraph of H ⊠ P for some graph H of treewidth at most c. This justifies that HUDGs are described as not having a grid-like structure in the literature, and is in contrast to unit disk graphs in the Euclidean plane, whose grid-like structure is evident from the fact that they are subgraphs of the strong product of two paths and a clique of constant size [Dvořák et al., '21, MATRIX Annals]. By allowing H to be any graph of constant treewidth instead of a path-like graph, we reject the possibility of a grid-like structure not merely by the maximum degree (which is unbounded for HUDGs) but due to their global structure. We complement this by showing that for every (sub-)constant r, HUDGs admit product structure, whereas the typical hyperbolic behavior is observed if r grows with the number of vertices. Our proof involves a family of n-vertex HUDGs with radius log n that has bounded clique number but unbounded treewidth, and one for which the ratio of treewidth and clique number is log n / log log n. Up to a log log n factor, this negatively answers a question raised by Bläsius et al. [SoCG '25] asking whether balanced separators of HUDGs with radius log n can be covered by less than log n cliques. Our results also imply that the local and layered tree-independence number of HUDGs are both unbounded, answering an open question of Dallard et al. [arXiv '25].