Despite what the video says about incognito mode in Chrome, private mode in Opera doesn't seem to cache favicons, not even if you bookmark the site. Entries for the sites and their favicons are not shown in the favicon file. (There are some exceptions in the file for default-provided speed dials though.)
As for clearing the favicon cache, I'm assuming Opera would just wait and inherit from Chromium whenever they decide to do something. But, it might be nice if Opera did like Brave. But, I think it'd be better to have an explicit option for favicons in the clear history dialog. Opera users love their favicons.
As for always fetching favicons fresh like Firefox does, that would be cool as a privacy option for Opera. Would be better if it was in Chromium though so other browsers could benefit from it.