Sigh, never mind. Google Play and the Opera site kept telling lies. With the tablet's default browser, I found a download page on the Opera site that said Opera was "Installing" rather than "Installed." It never finished, though. I managed to cancel it, and reinstalled from another Opera page that said, "If your download doesn't start right away, try going here:" or something to that effect. I went to this page, Opera installed after a while, and I had a proper icon.
Opera works just as expected, thanks.