I found a work-around that would probably help with other Google apps that don't work right with Opera. Apparently, Google deliberately makes it difficult for some apps to work with Opera since they want to force Opera (i.e. Google Chrome variant) users to switch to Chrome. (How nice of them!). I found a user-agent switcher add-on for Opera that is similar to the old Opera option that allowed Opera to be seen as another browser.
Here is the link: https://addons.opera.com/en/extensions/details/user-agent-switcher/?display=en

This add-on installs a toolbar button (similar to what ad-block does) that allows you to set Opera to mimic a wide variety of browsers. I added it to Opera Stable, opened Google Drive, and got the pop-up about using an unsupported browser, but was able to click on the X to close the pop-up, after which Google Drive looked fine!!

Yep! That's it.

Didn't work in Opera Stable

I closed Google Drive, then cleared browser cache, then reopened Opera Next and Google Drive. Square was gone! Will now see if that works in Opera Stable.

This morning the square is now showing up in Opera Next! Last night, it was just in Opera Stable.

Guess I'll use Opera Next for my default browser for now.

Opens fine in Internet Explorer.

It pops up every time I open Google Drive in Next or Developer, but I can click on the X to close it and the screen is then fine. I only get the black square with Opera Stable, which makes me think it is an error in how the error bar is being displayed (or trying to be displayed). There is no X to close it - just the square.

Just tried opening Drive again in Developer. I get the bar re: unsupported browser, but I can "X" it closed after which the Drive screen looks normal. I'm really beginning to think that the square I see in Stable ( and you see in Developer) is a malfunctioning unsupported browser bar.

Hmm. On my developer, it is fine. Maybe it is something unrelated to Opera itself. Since Google says that Drive isn't directly supported for Opera, maybe there is some other issue that is causing the problem. I do notice that when I open Drive in Next, there is a black bar on bottom left indicating that Opera isn't a supported browser, and the bar has a "X" to close it, after which the Google Drive screen looks fine. Maybe the black square is that same warning message not being displayed correctly?