Wikimedia Commons has this feature. Editors can manually bless certain combinations of traits as "subcategories".

For example, https://commons.wikimedia.org/wiki/Category:Paintings_of_cas... contains the subcategories "Paintings of castles by country" (nested hierarchy), "Frescos of castles" (a medium), "Paintings of Château de Chillon" (a subject), and "Young Knight in a Landscape by Carpaccio" (multiple views onto a specific item). Each item may appear in multiple subcategories. As far as I can tell, the UI won't let you search for frescos of Italian castles (unless somebody's made a subcategory for that), or view all paintings of castles regardless of their subcategory.

I'm not very fond of this approach. I'd prefer for each item to have an unstructured set of tags ("fresco", "depiction of a castle", "depiction of Italy"), with automatic derivation of parent tags ("fresco" implies "painting") and the option to search by multiple tags. It should be possible to automatically discover tags which best refine a search, so that the UI can still suggest them to the user, as it does today.