But they can control the content of the page. I looked at the source page and there is 0 mention of "search engine" on Google.com source page.

I'm pretty sure Google works contextually to know "search engine" means searching the web. Gogoduck does have the reference 5 times in the source code but Bing doesn't. Gogoduck can take the chance with their content to be the first, but if the leading search engines were to sue or act against Gogoduck, then it would be in Gogoduck's favor to get the spotlight.

If you go to Google to find Gogoduck then Google is of course the more valuable.