It might just be easier to tell them to clear their cookies and cache, and then try again. The amount of users this affects should be well below even 1% of the Twitch user base, just going on how often someone has come to me with this issue. Most people do not use multiple accounts on Twitch, and the ones that do may have bad intentions (ban evasions, for example).
I actually like how instant authentications are at the moment. Less time spent authenticating is a bonus to most users.