@Repter As I see on the extension support page in the Chrome Store, the developer knows about the problem in Opera. You need to wait for the next version of the extension in the Chrome Store or use the old version.
Funny thing, I tried the incognito mode and It didn't do anything, links were still opening in the same tab, even though none of the extensions were working at a time.
But then I tried turning off extensions one by one in standard mode and I found out that it was Custom New tab page extension that did this. I turned off two settings, even though they don't directly block new tabs, but it did the trick. Now left-click in a link opens it in a new tab. Just needed to restart Opera.
Thank you, Vux777, for your advice!