Other appsare terminated when I try to copy an auth code from the Google Authenticator
You can't use an app specific passphrase. Stackexchange uses Google as an OpenID provider and that works with access tokens.

To continue/finish the OpenID process you need to be logged in to Google of course (using the regular 2-step passphrase+token method). You will then be asked if you want to grant access by Google and the app will get an access token.