Hi there!
When you acquire any Storage Chest furnishing, it will be unlocked in the 'Furnishings' tab of your Collections. You can place these collection furnishings once in every home you own each, provided that you have a free 'Collectible Furnishings' slot in the home. These slots are shared with Undaunted Trophies and many other achievement furniture such as the Vvardenfell model or the Augur of the Obscure, etc.
So make sure that you haven't reached the 'Collectible Furnishings' slot limit for your house (open the editor, go into browse, and check the displayed numbers). Even in the smallest house type (Inn room), you should have 1 Collectible Furnishing slot (or 2 if you have ESO+ which doubles your furnishing limits). So if you really have no other collectible furniture in that house, and still can't place the storage chest, that's a bug.
You can use the Retrieve function (open editor, go into browse, switch to the Retrieve tab on the top right) to make sure that you don't accidentally have some furniture placed outside the walls or something where you might not be able to see them at first glance.
But normally, you can place the same storage chest in all your houses from the editor's Place --> Services --> Storage menu.