The microphone on Duolingo is only meant for Chrome and only works in Chrome. The Speech-to-Text API that's used uses a Google service for converting the speech to text that requires special API keys from Google that are only allowed in Chrome. Google won't let other Chromium browsers use the service.
For this to work in Opera, Opera would have to modify the Chromium source used for Opera and add its own Speech-to-Text API engine that used a non-Google service (which would cost money if Opera didn't do it in-house).