Sounds like PuTTY may be sending extra line feeds/carriage returns to the Digi. The 2nd Login prompt says that you didn't (technically) enter anything on the first one, so it's showing it again. The skipping of the password prompt is like you pressed enter twice after typing in your login name.
You can check the connection settings to see if there's anything relating to an extra line feed (^M), and uncheck or disable it and try again.
You can check the connection settings to see if there's anything relating to an extra line feed (^M), and uncheck or disable it and try again.