Old Opera used to have each tab being a separate environment. For some reason we are mostly back to "private windows" now, which aren't separate at all between each other.
We changed that for a reason. No use wants to be logged out that often. Actually most facebook user probably enter their password one time a month. Less if they use the mobile.
Firefox is integrating a cookie feature from Tor called first-party isolation or double-key cookies. It will separate third-party cookies for each first-party site. If a.com and b.com both load images from evilcorp.com, Firefox will send evilcorp.com different cookies for requests from a.com and b.com. Blocking third-party cookies can break some site that rely on third-party resources, but first-party isolation should allow each site to work without cookie "crosstalk".
You can test first-party isolation now by flipping the about:config pref "privacy.firstparty.isolate" to true. Beware that there are still bugs that break some sites, which is why the feature is not enabled by default yet. If you find bugs, please report them in Bugzilla! Here is the Firefox bug tracking the integration and known bugs:
Thanks for the tip, I didn't know about that. I'll play with that this weekend. (Definitely not afraid of breaking sites; that's how I learn what they're up to.)