Recent observations indicating the existence of a monochromatic gamma-ray line with energy ~130 GeV in the Fermi-LAT data have attracted great interest due to the possibility that the line feature stems from the annihilation of dark matter particles. Many studies examining the robustness of the putative line-signal have concentrated on its spectral attributes. Here, we study the morphological features of the gamma-ray line photons, which can be used to differentiate a putative dark matter signal from astrophysical backgrounds or instrumental artifacts. Photons stemming from dark matter annihilation will produce events tracing a specific morphology, with a statistical clustering that can be calculated based on models of the dark matter density profile in the inner Galaxy. We apply the DBSCAN clustering algorithm to Fermi gamma-ray data, and show that we can rule out the possibility that 1 (2, 4) or fewer point-like sources produce the observed morphology for the line photons at a 99% (95%, 90%) confidence level. Our study strongly disfavors the main astrophysical background envisioned to produce a line feature at energies above 100 GeV: cold pulsar winds. It is highly unlikely that 4 or more such objects have exactly the same monochromatic cosmic-ray energy needed to produce a gamma-ray line, to within instrumental energy resolution. Furthermore, we show that the larger photon statistics expected with Air Cherenkov Telescopes such as H.E.S.S.-II will allow for extraordinarily stringent morphological tests of the origin of the "line photons".