From my understanding, the reason Chromium works fine is because it now ships with its own multimedia libraries. Opera can't include those libraries due to licensing issues, and since Opera is based on Chromium, and Chromium now has its own libraries (and no longer supports using system libraries since there's no need to), Opera is stuck with not having any built-in libraries and not supporting any external ones. (Is this correct? It's just what I pieced together from blog comments and forum posts.)
I'd also like to add that for some reason, YouTube still seems to think that Opera has H.264 support (see youtube.com/html5), which leads to video playback errors when it tries to serve MP4 videos. YouTube still works fine when the videos are served in WebM. Eventually we'll want these issues resolved, but for the time being, is there at least a way to tell YouTube that my browser doesn't support H.264?