For a very, very long time I haven't seen pages where you need to mask a different browser in order to display them correctly. Manage My Kindle hasn't been working in Opera browser for a long time. I thought the trouble was the Opera browser and thus I was using different browsers. But it wasn't ...
The problem isn't in Opera being unable to display the content. Rather it is in browser detection which then modifies the content. For Opera, however, the content is not loaded. For details see Resource - I haven't explored it.
Edit site preferences, Network tab, Mask Opera as Internet Explorer. WOW.