Just want to chime in on this. I just had this same issue last night, so the problem is still present in the current builds (Opera mobile 49.2 and beta 50.0) both for Android. I spent several hours last night trying to figure out why I couldn't log in or sync my browser settings, after I changed my password recently. Finally stumbled on this post. It's a pretty big design flaw that one cannot participate in the forums for help without logging in... Lol. The steps provided by the OP above WORKED! So if anyone has this problem just follow these steps and it should fix the issue. I'll repeat them here:
"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."