This problem is to be put in the context of browser html5 support, more specifically support for h264 streams (which affected much more than Youtube live streams). That seems to be lacking at the date of these posts in both Chromium and Opera but can be solved only separately.
For Chromium:
sudo apt-get install chromium-codecs-ffmpeg-extra
which will remove chromium-codecs-ffmpeg
if already installed.
For Opera:
- Go to opera://about/ and look for the relevant Chromium version.

Download from here the corresponding version of chromium-codecs-ffmpeg-extra
Extract the archive.

- Copy the
libffmpeg.so
file (with the file manager as administrator or in terminal) to /usr/lib/x86_64-linux-gnu/opera
replacing the already existing file if needed.
(This is different from a solution on which it is loosely based. Location is different and no renaming of the file was done.)
This was first answered here, also here.
The libffmpeg.so
file can be also found in the package opera-ffmpeg-codecs
here and here, and it can be also put in a new folder /usr/lib/x86_64-linux-gnu/opera/lib_extra/
.
The above may need be repeated depending whether updates replace the libffmpeg.so
file or require a new one corresponding to a newer Chrome version.