SOLVED
I have an answer from a Vivaldi ambassador on the Vivaldi forum which I will quote in full since, although it makes clear that the issue lies in Vivaldi, it does relate to Classicpress and its use of cookies. The ambassador installed Classicpress on a test site to experiment and found this:
I was able to reproduce the problem on my test site.
Found a related bug report: VB-107557
Looks like it only happens if the link is opened from the context menu.
Technically, cookies with SameSite=Strict are not sent in new tabs opened from the context menu. This is not limited to ClassicPress.
I will ask the devs to prioritize this one, it’s a nasty one.
For now, don’t use the context menu to open links.
Use Ctrl-click or Shift-click - it’s also faster 
That’s why I couldn’t reproduce this, I always use Ctrl/Shift-click.
You did say “right-click on a button” but I wrongly assumed it would behave the same… 
It also looks like WordPress does not use SameSite=Strict cookies.