When Firefox browser was the default browser in Windows 10, with Cortana enabled, the web searches from search box will open in Mozilla browser with the default search engine. Things changed recently, Cortana search no longer works with Firefox.
The reason is this: Microsoft has found their digital assistant searches being redirected to other search providers by browsers such as Firefox and Chrome. the software giant has blocked those browsers from using Cortana and made the Windows 10 search experience to only function with Edge browser, powered by Bing. Windows 10 Anniversary Update includes this change. This means, you can no longer do Cortana searches with Firefox, so as the case with Chrome also.
After realizing this recently, Mozilla has removed the redundant Cortana-related code, this also removed the option ‘Use search engine for searches from Windows’ from Search settings. You can spot this change in Firefox 52.