Yes, but you can't add a new search engine at all! So if a search engine isn't one of the tiny number of options in that dropdown, you can't change to it. That applies on both iOS and MacOS. And that option is used for the entire system-wide search, not just Safari.

So here's a challenge, try adding a search engine not on that list. You can see the search engine I'm working on in my profile if you're interested (I don't want to hijack this thread with self-promotion). I challenge you to change to a new competitive option like it. You simply can't. That is a clear monopoly over distribution.

On desktop in Chrome, as noted it is not something any typical consumer can do easily. But even if they could, Google does not allow you to set the New Tab to another search engine, even by setting the homepage to one. So every new tab opened on Chrome takes you back to Google search, even if a consumer figures out how to change their homepage. As for changing the nav bar search, no ordinary consumer is going to be able to work out how to change a search URL pattern. That is clearly intended to prevent consumers changing.

So I stand by my point, especially on an iPhone, you simply cannot change your search engine to a new search engine like us. It is impossible.