@jimunderscorep,
Thanks for verifying the site works for you, with Opera. With that bit of information, I continued to work on another possibility, and that is, there may be an incompatibility in the selection of Extensions installed on this copy of Opera.

I clicked 'Disable' on all of them and tried transfer.pw again.

Now it works normally, for the first time!

I'll have to bring the extensions back again, one by one, and see which is'killing'that web site.

Thanks also for letting me know there is a more recent version of Opera. I wouldn't have suspected that, with About Opera reporting 'your system is up to date' -- when that's not the case. Also, Synaptic fails to show there's a version 46.x.

If we may off topic a bit, which repo is giving you 46.x? Obviously, the ones I'm using don't have it.

The transfer.pw site also works correctly with Vivaldi and Firefox -- in addition to Chromium.

On Opera, since the original post was made, I've been working with changing the User Agent string, but either I'm not doing it right, or it has no effect because the problem lies elsewhere.

Open this web page and type any text into the box with the 'Lock Icon'.
Then, click the Encrypt and Transfer button.
This takes you to the next page, where you should be shown a link where the unencrypted text is displayed.
In Opera, the link is replaced with an error message.

Do the same in a different browser that supports (as does Opera) HTML5 sessionStorage. For example, in Cromium, the site behaves normally, with a link being displayed.

Is this just for me, or can anyone else duplicate the problem?

Incidentally, I've just filed a bug report on it, since it happens repeatedly and only on my Opera installation.

Version information:
Version: 45.0.2552.812 - Opera is up to date
Update stream: Stable
System: Ubuntu 14.04.4 LTS (x86; KDE)

Browser identification:
Mozilla/5.0 (X11; Linux i686) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/58.0.3029.81 Safari/537.36 OPR/45.0.2552.812

With Opera v37:
Main Menu -> Speed Dial -> Bookmarks tab opens -> Other Speed Dials

Open the last one and there is a folder with the name of the remote computer that's exporting its Speed Dials over the sync connection. I expect there would be more than one line, if one is syncing multiple devices.

Keep checking the sync connection. For some reason it dropped, and then I was wondering why test folders on one device weren't appearing on the other. Connection re-established, function resumes.

It's all working, otherwise, as kelleric9 writes above. Yes, there is a learning curve, but after that it gets better.

After following this thread, I feel most fortunate to have started with v37 ...