Previous searches for the γ-ray signatures of annihilating galactic dark matter used predefined spatial templates to describe the background of γ-ray emission from astrophysical processes like cosmic ray interactions. In this work, we aim to establish an alternative approach, in which the astrophysical components are identified solely by their spectral and morphological properties. To this end, we adopt the recent reconstruction of the diffuse γ-ray sky from Fermi data by the D3PO algorithm and the fact that more than 90% of its flux can be represented by only two spectral components, resulting form the dense and dilute interstellar medium. Under these presumptions, we confirm the reported DM annihilation-like signal in the inner Galaxy and derive upper limits for dark matter annihilation cross sections. We investigate whether the DM signal could be a residual of the simplified modeling of astrophysical emission by inspecting the morphology of the regions, which favor a dark matter component. The central galactic region favors strongest for such a component with the expected spherically symmetric and radially declining profile. However, astrophysical structures, in particular sky regions which seem to host most of the dilute interstellar medium, obviously would benefit from a DM annihilation-like component {as well}. Although these regions do not drive the fit, they warn that a more detailed understanding of astrophysical γ-ray emitting processes in the galactic center region are necessary before definite claims about a DM annihilation signal can be made. The regions off the Galactic plane actually disfavor the best fit DM annihilation cross section from the inner Galactic region unless the radial decline of the Galactic DM density profile in the outer regions is significantly steeper than that usually assumed.
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