I've never actually run a consignment store. I think there maybe mods that help with actually running the shop over at Nraas and MtS.
As far as I understand it the register does most if not all the consignment stuff, meaning handles the stock. You can browse/buy/sell by clicking on the register and going through the options.
You don't have to own the shop for the register to do it's thing.
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Here's what I've done to build a Consignment Shop: (Finding the Register)
1. Picked a lot and change it to a community lot with a type of Consignment Store. Go into lot.
2. In build mode, click on community objects (it's the building on the right of the build menu)
3. Search by Ambitions by using the filter content button on the right of the menu.
4. Click the Misc. Objects tab (2nd tab) to find the consignment register.
-- If you'd rather want the Elixir/Consignment Register, filter content by Supernatural and it should show up. Elixir Store Consignment Register. This is the one I usually like using due to it allowing to be open all hours day and night.

You can have both registers on the lot to make it a combo lot. The game gives both registers NPC to run the registers.