User Authentification on consoles

Suggests that you are creating a game on console.

My suggestion is that your Game must have a way to talk to ConsoleServices and see the gamertag/whatever.

Create a website for you game that allows you to login via ConsoleServices and then link the twitch to that.

Or create website that logins via Twitch and then displays a “link code”, enter the link code into your game on the console