In Opera GX only, you can goto the URL opera://settings/early_bird, turn on the Guest Tabs" feature and restart Opera. Then, when you're on a site, you can right-click the tab and choose to open that site in a guest tab to sign into that site with a different account. Guest tabs work in a normal window. No need to use a private window.
The downside Guest Tabs though is that all guest tabs share the same session. This means the feature will only allow you to log into a site one 1 extra account.
With that said though, it shows that Opera might be able to hack something in themselves. It might just take a lot of work.
It really is a bummer that in Chromium, all tabs in all private windows share the same session. Would be nice if Chromium could do something about that.
I think Zen on the other hand does something cool. You can create extra workspace and then set each workspace to use a different profile (and therefore session). That might be nice. too.