I believe the "flat" was meant in contrast to "curved", as in typical fisheye/ultra-wide-angle lenses.
> the new fisheye lens consists of a single flat, millimeter-thin piece of glass covered on one side with tiny structures that precisely scatter incoming light to produce panoramic images.
With a thinness of a millimeter, I'd say that's flat. The "tiny structures" are in the form of a film, which further qualifies as flat.
> Their new metalens is a single transparent piece made from calcium fluoride with a thin film of lead telluride deposited on one side. The team then used lithographic techniques to carve a pattern of optical structures into the film.
> Each structure, or “meta-atom,” as the team refers to them, is shaped into one of several nanoscale geometries, such as a rectangular or a bone-shaped configuration, that refracts light in a specific way.
Click bait. Not "completely flat" at all.