Hi Jan, a few things to note: the color management color profiles cascade—if a placed image has no color profile assignment (an image listed as Document RGB or CMYK in the Link Info panel), the InDesign document’s assigned profile (Edit>Assign Profiles...) manages the preview. If the InDesign document has no profile assignment, the fall back for the preview is the Color Settings’ current Working Color Space. In Photoshop the fall back for an image with no embedded profile is also the current Color Settings.
Your capture of the InDesign Links panel shows a placed PNG as DocumentRGB (no Embedded profile). Your capture of InDesign’s Assign Profiles... dialog also shows no profile is assigned to the ID document, so the preview of the PNG is falling back to your current Color Settings’ Working RGB profile.
Your last post shows a capture of an image in Photoshop (z-PhotoShop-StatusBar-01.jpg) in CMYK mode with Coated Fogra 39 assigned to the image. You can’t save PNGs as CMYK, and if you Save As PNG, the color will be converted into RGB on the save—the save dialog lets you choose whether to embed a profile.
If you save the CMYK file as a .PSD with no profile embedded and place it into an InDesign document with no profile, it will get the current Color Settings’ CMYK Working profile, which is US Web Coated SWOP in your first set of captures. So, in Photoshop the image preview is Coated FOGRA 39, but in InDesign the preview would be US Web Coated SWOP, which would show a different color preview.