As long as you see the "DuckDuckGo Suggestions" list, you know which search engine you are using. I don't even know how to get a list of icons to show in Safari, although I do see that in iOS. Perhaps that is the "Favorites Bar"? If so, that explains the text "clicking on the icon does not indicate duckduckgo.png".
I had no idea what a duckduckgo.png had to do with anything.
I now assume you were talking about clicking the
duckduckgo icon in the Favorites list. I would assume that that icon would simply take you to their site. As far as I know, those Favorite icons simply take you to whatever site they represent, they are not links to a search engine.
You answered/posted while I was posting the above.
Not only do I not use the "Favorites" list/bar, I never use any browser's bookmarks list. I have used a completely separate bookmarks manager for well over a decade. It gives me the same list of bookmarks, in any major browser, and negates any need to collect/sync/save/move/etc. them as my browser-of-the-moment changes.