Update: I was able to fix the issue of Opera for Android asking me to enter my Opera password as a "passphrase" by:
Opening Opera desktop and choosing Settings > Advanced synchronization settings > Reset synchronization. I logged back into Opera desktop with the same username and password as before. Then I logged into Opera Beta for Android and the "Passphrase needed" error was gone.
It seems like the Reset synchronization option needs to be added to Opera Beta for Android to allow Android users to fix this issue without having to have a PC or Opera desktop.