In your Finder example above, Finder isn’t showing the app icon, it’s showing a preview of the file.
(There is a setting in DT for showing the file preview as a thumbnail instead displaying an app icon, but you haven’t asked about that.)
In Finder, if you right click on a markdown file and click “Get Info”, you should see this screen. In the top left is the icon thumbnail, then the second circle in my screengrab shows the default app and the third button sets it for all markdown apps.
Assuming the last two are set correctly (your default app and it’s set to all), the problem I then had was that the app image in the top left didn’t update after I’d changed the default file type. (I am one of the many posts on this forum about this problem!) I updated the icon myself by dragging the correct image to the field, and after that it was fine.
But I did change the setting in DT to show the file preview anyway, since that’s the behaviour I actually want!