On the April ESO Live, ZOS explained why they can't during the Housing Q&A. Basically, it gets more complicated than we'd think when dealing with items that don't stack up into limited categories.
Or the ability to deconstruct and get mats back if I can’t store the items themselves
Absolutely!
I have been hoping they'd give us something like a crafting bag for furnishings since housing was released.
Even with the option to store them in other homes, banks, or storage chests.. they just can't keep up and "house" all of the furnishings they continue adding to the game.