If you use Firefox Containers, including the Facebook Container, please also use Cookie AutoDelete [1] to get rid of cookies from closed tabs across containers. Otherwise, in my observation, sites will still be able to track you if you reuse a container (even after closing all tabs of that container) for a specific site.
I know the purpose of containers. But it's still not as easy to use as I would like it to be (I understand that doing usability design for this is complex), and I haven't created a specific workflow for it yet.
When I'm on some site and want to quickly search the web using Google (when DDG isn't enough) on the same tab and then return back to the site, or if I use a container temporarily to check a Gmail account, I don't want to mix those up by mistake later (I never search on Google while logged in). This problem may not exist if I spent time to pre-decide the use/assignment of websites/accounts for each container.
Since I also use sites like HN to jump to other sites (several of which may be trying to track), auto-deleting cookies gives me peace of mind once I close a tab (I set the timeout value in the extension configuration quite low). I was using Self-destruct Cookies [1] before (when legacy addons were still supported on Firefox).
For additional protection, I do some more extensions.
The best feature (at least for me) is that you can whitelist some specific websites to always open in a new temporary container, without any extra effort, while leaving the rest of your websites like they are today.
For example, I have YouTube (and any other Google website) to always open on a temporary container but other sites like HN and Reddit are on a normal and permanent container.
I believe Incognito disables (some?) addons, and it has a collection of settings which may mean it acts differently from regular browsing (Tracking Protection is probably the biggest, although I turned that on for normal browsing too).
It's also unclear to me where one Incognito session stops and the next starts, although I'd assume they live and die with their window.
More importantly, Incognito opens a new window while Temporary Containers open in the same window with coloured tabs. So you can have _lots_ in the same window if you want.
My understanding is that First Party Isolation maintains _separate_ caches per first party, but I don't know whether temporary (or multi-account- for that matter) containers also do that. It seems like it would be reasonable for them to do so but I can't find a reference.
For some reason Firefox Containers with Cookie AutoDelete messes up with Twitter and I cannot login to Twitter. After entering user/pass I expect 2FA screen but I'm redirected to `https://twitter.com/login/error?username_or_email=...` url where I have to enter my credentials again. I'm not sure if I'm the only one.
I don't use 2FA with Twitter, but I have no issues logging in with Containers and Cookie AutoDelete. Just a data point that something else may be amiss on your end. Disabling other extensions and trying it could possibly help narrow down on the cause.
[1]: https://addons.mozilla.org/en-US/firefox/addon/cookie-autode...