The motifs in the crown store don't give any housing furnishings.

They simply unlock the crafting styles as if you had read all X pages of that motif type on the character which uses it (they're not account wide unlocks).
From there you can craft or use the outfit system for that style.

It's worth noting that most motifs sold in the store are very bad prices vs what they're worth for gold in game. Buoyant Armiger might be one of the few exceptions but even that's fallen in price now due to the event.