That is because the native Affinity file format is proprietary & can only be opened by one of the Affinity apps (Designer or Photo & eventually Publisher). As Patrick said, you can export Affinity files to other formats that other apps can read, but only the Affinity apps support certain features like saving the history with the file or snapshots. The logo on Affinity format document icons are just an indicator of which Affinity app will open it if you double-click on them but they do not appear in the document itself or in files exported to other formats.
The native files loaded and saved belong to the application and have an icon as you describe. Files can also be exported as JPG and other common file formats.
You might want to look at the export section of the help or the exporting video
