The distinction you are making doesn't exist. localStorage and cookies both persist after the browser is closed so if you want that use case you won't get it from either.
Maybe they were thinking of the options most browsers have to clear selected private data on exit. In that case, though, it's still possible to clear both localStorage and cookies I am sure. Maybe parent commenter currently has the options set so that cookies are cleared but localStorage is not and that's why they thought that this is the way it always is.
