Is it possible to fetch the User nickname from App Store account information?
App Store Nickname
Where is this nickname defined ?
To retrieve the nickname of user who write reviews on App Store.
I don't think so. That's one of the goals of nicknames, to protect identity of writer. I guess only Apple could find it.
So you have to answer to the review and politely ask for the author name or mail…
App Store Nicknames are set when a user with an Apple ID writes their first review within the App Store. Changes to a chosen Nickname can be made with the Apple iTunes software.