From the site, it looks like the FB login page you get after clicking the button is all black, and then you get sent back to the app. The regular one would be in the normal colors. Although I suppose they could popup a fake anon one on their own site, but they wouldn't have the right URL. On Android the app allows login without entering a password as well, so I'd immediately know because login through FB doesn't require my password but the app's fake version would.
the question was about posing non anonymous FB login as anonymous to trick users into giving more info about themselves. neither should require a password if yo're already logged into FB.
