The only way to kind of achieve this is by doing a global search with a prefix, like 'item: widget' and pressing Enter so it brings up the results of an item search, which is customizable.
(The un-prefixed global search results page is not customizable; the only way you end up in the filtered results screen is if the only match for whatever you searched is one single record type - and this might be why you and your user see different things on the search, different permissions, different found results.)
If you see on your screen 'Global Search: Item Results" then there should be a Customize button under it, and you could include Quantity Available in that.