@pindos

Flash Disabled by Default (Target: Chrome 76+ - July 2019)
Summary
Flash will be disabled by default, but can be enabled in Settings at which point explicit permission is still required for each site

https://www.chromium.org/flash-roadmap

You will need to allow Flash on sites every time you start Opera.

This hasn't been my experience. When I first started poking around forums looking for a solution, the question of whether flash was enabled or not came up. When I checked on it, I saw that it was already enabled and I had never fooled with it before. And it has remained that way with no sites excluded ever since, except when I explicitly disabled it while experimenting. Possibly that is a characteristic unique to the rpm distribution.

If the last suggestion hadn't worked, I was going to uninstall Opera and try the snap version. It would be interesting to see if it has the same problem.

The thing that puzzles me in all of this discussion is why it's even necessary to go through all of this to enable a feature which is used all over the Internet. It looks like the Opera developers need to go to work on this one.

Flash Disabled by Default (Target: Chrome 76+ - July 2019)
Summary
Flash will be disabled by default, but can be enabled in Settings at which point explicit permission is still required for each site

https://www.chromium.org/flash-roadmap

You will need to allow Flash on sites every time you start Opera.

About the substitution of that "bad" libffmpeg with a "good" one.
As I have before, solutions like these should start with a "proceed at your own risk", because you are replacing something that the app needs to work, with something else, built differently.
There was one thread some months ago where some debian user's opera crashed at launch because of that "good" libffmpeg he changed.

As for opera forgetting sites that you have allowed flash to run.
If I remember correctly, this is a feature of chromium 75 or so and later. Sites are allowed to run flash as long as the browser is running. Once it is closed, the browser forgets it.

For example, you can download one of the chromium-codecs-ffmpeg_77.0.3865.120 packages for x64 arch from here, e.g. this one, and unpack the "libffmpeg.so" file to "/usr/lib/chromium-browser/" folder, or if you don't want it be overwritten by some other version of that lib, you can copy it to "Opera's-installation-root/lib_extra" subfolder (you need to add it yourself). You can find the path of installation root in opera:about > Paths > Install (the default in your case should be /usr/lib64/opera).

If it still doesn't work, try to run the opera with the switches as show below:
/usr/bin/opera --ppapi-flash-path=/usr/lib/adobe-flashplugin/libpepflashplayer.so --ppapi-flash-version=32.0.0.270
and see if this helps.

If BBC is falling back to something else, I would have to assume that it's not WebM because when I disable Flash in Opera, I always get a notification that I have to install Flash to view the video.

When I went to Opera's site to download the program, I noticed that it is definitely Ubuntu-oriented. I had to scrounge a bit to find the rpm version so I'm guessing that they have made a corporate decision to focus on Ubuntu and derivatives and possibly I will have to finally abandon Opera. It's not that I like Fedora better than Ubuntu, only that for the present Fedora is the only flavor of Linux that actually works right on my computer, at least of the 4 or 5 distros I've tried.

As for bbc's site, it seems that it uses flash as a fallback option, like some other sites do. How did i find out? It plays on firefox and chromium, where I keep flash disabled.

Regarding h264. Opera does not use gstreamer's library for h264 (maybe the presto opera did, but it is as dead as internet explorer). Opera uses its own libffmpeg, which does not support h264. And no matter what you may install, unless it is that libffmpeg from chromium ffmpeg codecs on ubuntu, it still does not use them.

Last but not least, youtube (and only youtube) uses webm as a fallback for browsers like opera that do not support h264. Webm has several drawbacks compared to h264, e.g. lack of hardware acceleration.

When I went to the Adobe link you suggested, it told me to enable Flash and gave me a button to do so, even though I've checked it two or three times in Opera's settings and it was always enabled. After clicking on that button the flash video worked, but when I went back to Aljazeera, it still wouldn't work. So I tried a video on bbc.co.uk/news and got the message about enabling flash player again. So I clicked again and the video played. At the same time they showed a message saying I would have to upgrade my browser or device because in 2020 Flash would be history.

Now, regarding H.264, Fedora does have GStreamer H.264 codec, which I installed, though I have no clue as to whether Opera can deal with it or not. After reading your message I poked around until I discovered that I could also install ffmpeg via the rpm fusion repository, so I did that too. And just to be sure, I then restarted my computer. It was after restarting that I did the above test at BBC and Aljazeera. So there's something else going on here and I guess that until someone figures it out, I'll have to use Chrome, at least for sites with videos that I want to see.

Incidentally, I never have problems with youtube videos, which of course are html5

Second, the sites probably use h264 and opera does not support it. For the why, check the post on my signature and use another browser for these sites, because fedora does not package chromium ffmpeg codecs.

Third, as fas as flash is concerned, check it on a page that uses a flash element, e.g flash's about page here, and do not rely on sites that use javascript to detect it, e.g the one mentioned above.
https://get.adobe.com/flashplayer/about/

Trick question :p
Are you using the snap or the deb version of opera? If you do not know the answer, tell us how you installed it.

/usr/lib/adobe-flashplugin/libpepflashplayer.so

Yes, file is there

If you have it on a different path, use dpkg -S to tell you which package provided the file, e.g.
$ dpkg -S /usr/lib/adobe-flashplugin/libpepflashplayer.so flashplayer-chromium: /usr/lib/adobe-flashplugin/libpepflashplayer.so

(I am on debian, so the file is provided by package flashplayer-chromium from dmo repo)

Adobe's page launches apt and installs adobe-flashplugin from ubuntu's partner repo, if you select the first option, aka "apt for debian/ubuntu", so the forementioned file should be on the right place.

Do you have the correct Flash installed? Did you added the site to the exception list?