Or rather, you can, but it loses the name when you pick it up and re-place it.

I used an anvil to name my Luggage, but it didn't work. :(

Does the name even show in the gui?

It does. Renaming a Luggage item with an anvil to Foo works; you end up with an item called Foo with the Luggage icon and item ID.

If you then place the item in the world, the tooltip doesn't have the new name, and if you pick it back up, you're back to an item called Luggage again.