Vision-language models have achieved remarkable success in multi-modal representation learning from large-scale pairs of visual scenes and linguistic descriptions. However, they still struggle to simultaneously express two distinct types of semantic structures: the hierarchy within a concept family (e.g., dog $\preceq$ mammal $\preceq$ animal) and the compositionality across different concept families (e.g., "a dog in a car" $\preceq$ dog, car). Recent works have addressed this challenge by employing hyperbolic space, which efficiently captures tree-like hierarchy, yet its suitability for representing compositionality remains unclear. To resolve this dilemma, we propose PHyCLIP, which employs an $\ell_1$-Product metric on a Cartesian product of Hyperbolic factors. With our design, intra-family hierarchies emerge within individual hyperbolic factors, and cross-family composition is captured by the $\ell_1$-product metric, analogous to a Boolean algebra. Experiments on zero-shot classification, retrieval, hierarchical classification, and compositional understanding tasks demonstrate that PHyCLIP outperforms existing single-space approaches and offers more interpretable structures in the embedding space.