Is it possible to authenticate into Stack Exchange using this method? At the moment there's other methods such as OpenID, Google, Yahoo and Facebook, but it would be interesting to also have the GitHub authentication.
You can log into any Stack Exchange site using the credentials from any Open ID provider. Unfortunately, at this time, GitHub is not an Open ID provider, so you can't log in using those credentials.
Also, it's highly unlikely that SE would create a special login mechanism for GitHub accounts unless there was great incentive to do so; it would require resources (time, money) to fabricate a custom mechanism when one already exists for this specific purpose (Open ID).
If anything, I suggest that you urge GitHub to become an Open ID provider, rallying support around that initiative. Once they are an Open ID provider, you'll be able to login to any Stack Exchange site with those credentials and link them to your account.
The PR was finally approved and merged: https://stackoverflow.blog/2019/11/11/announcing-support-for-github-authentication-in-stack-overflow/