No, I do not think it's fair to say that Apple allows other browsers, and neither does the DOJ.
This is the correct take, though. They ARE the same web browser, just with different skins.
This isn't a sufficient description. Apple actually requires all third-party browsers to use the WebKit framework. If they actually allowed browsers to use the WebKit engine, then you could make a new browser incorporating the open source WebKit engine compiled into it. But this is not allowed.
> Saying there is only one browser on iOS is like saying Google Chrome and Microsoft Edge are the same browser because they both use chromium.
No, Edge and Chrome both use the Chromium engine. They are different browsers that incorporate the same engine. Edge can do this because Chromium is open source.
But Chrome on iOS doesn't have its own engine at all (not even the Webkit engine). It just offloads to the WebKit framework.
You cannot for iOS. On iOS, you have to use the WebKit framework. Your app will not be compiled with any WebKit open source code.