When I installed Firefox yesterday, I immediately set my default search engine to DuckDuckGo and removed Yahoo from the list. Today, I discovered that Firefox has a search provider built in for when you swipe up from the home button. I decided to try it out, but noticed that the search engine was set to Yahoo there. The settings only had an option to clear search history. I also tried tapping the Yahoo logo and the search icon, but it only started a search. Is it possible to change the search engine in this thing? Or is the Mozilla-Yahoo deal that far-reaching? Images:
If you tap on one of the search engine, it should give you this dialog box which you can tap on "Set as default":
I tested it on a Nexus 6 and it does switch the search provider used in the Home-button swipe search.
If this dialog doesn't appear for you or if the "Set as default" here doesn't change your search in both the URL bar and Home Button search, then it would be a bug. You might want to report the bug in Bugzilla@Mozilla, with your device and OS version.