Abstract: In this work, we demonstrate how the composition of two unsupervised clustering algorithms, AstroLink and FuzzyCat, makes for a powerful tool when studying galaxy formation and evolution. AstroLink is a general-purpose astrophysical clustering algorithm built for extracting meaningful hierarchical structure from point-cloud data defined over any feature space, while FuzzyCat is a generalised soft-clustering algorithm that propagates the dynamical effects of underlying data processes into a fuzzy hierarchy of stable fuzzy clusters. Their composition, FuzzyCat $\circ$ AstroLink, can therefore identify a fuzzy hierarchy of astrophysically- and statistically-significant fuzzy clusters in any-dimensionality data set whose representation is subject to changes caused by underlying processes such as stochasticity and temporal evolution. Furthermore, our pipeline does not rely on strong assumptions about the data or the number/importance of specific structure types, allowing it to produce these outputs without requiring much input from the user -- thereby making itself applicable to a wide range of fields in the physical sciences. We find that for the task of structurally decomposing simulated galaxies into their constituents, our context-agnostic approach has a substantial impact on the diversity and completeness of the structures extracted as well as on their relationship within the broader galactic structural hierarchy -- as among the structures extracted with an out-of-the-box application of FuzzyCat $\circ$ AstroLink are; dwarf galaxies, infalling groups, stellar streams (and their progenitors), stellar shells, galactic bulges, and star-forming regions.