It would be odd for Google to invest in ranking high in Google searches for "search engine" and I wouldn't be surprised if they actively suppress that result because -- well -- people searching on Google have already found Google's search engine. But Google does land on the first page of Bing results for "search engine".
In any event, I'm not sure how this relates to my question. Can you help me understand? For example, do you think Google -- who most would agree are a fairly important search engine -- are trying to rank high on Google searches for "search engine" and discovering that for unknown reasons, they can't even get into the top 15 pages of results for it? That is what I am asking about, except it's Firefox and "browser".
Firefox is second for "browser" on Bing. It's been a browser for twice as long as Chrome has existed. It is at least as well optimized for the keyword "browser" as, say, a random GitHub project on the 14th page of "browser" results. And yet, it doesn't rank. I'm looking for ideas why.
For what it's worth, unlike Firefox, Google may not need to invest in search rank for, say, their browser. Anyone who visits Google Search (or any other Google tool) in a non-Chrome browser is apt to get an advertisement at the very top of the page, outside and above all search results, encouraging them to use Chrome. You don't need a lot of SEO when you own the SE.