Ah, just as I post my method of creating a shortcut with an icon of my choice on the iPhone home screen - which I demonstrated with Firefox - along comes a browser that allows a choice of icons out-of-the-box!

What timing! DuckDuckGo Privacy Browser starting with version 7.37.0, released on 31 Jan 2020, has a feature to customize the home screen app icon! Just select the background colour of your choice from Settings > App Icon:

DuckDuckGo Privacy Browser App Icon customization

So now I can satisfy my OCD:

iPhone Dock with DuckDuckGo Privacy Browser

Nice! Can more apps provide this? Hint: green background please.