I use two-step verification for Dropbox - and my verification device is my phone, so it's not possible to open Aldiko Sync, login to Dropbox (twice), then go to my Google Authenticator app for the code, and come back to Aldiko Sync - the authentication process is interrupted as soon as I switch to the Google Authenticator app.
And there's not enough time to write down the code from Google Authenticator, and then go to Aldiko Sync (believe me I tried several times).
What do you suggest to authenticate to Dropbox for me? I'm not sure it is an option, but other apps handle authentication a little more "gently" (such as Dropsync), authenticating as a Dropbox app instead of using password authentication. I can't recall exactly how the process was different, but I was able to authenticate Dropsync without a problem.
You can try disabling two factor authentication, allow the sync app to login and then turn it back on.
From what I've heard that works reliably.
Please let me know if that didn't work for you.