If all you want to do is separation of data on same site. I think the container is the better way to go.

Although the profile still serves different purpose from container (like you can have different Firefox account linked/ different extension installed)

I actually want the kind of UI chromium provides, where I can see from the background color directly in which "mode" I'm in. This makes it easier to separate work from non-work.

That's exactly what Firefox containers do. I have a different color for each one.

I've used containers and I feel what they do is not exactly what I want. I actually want a separate instance of the whole browser.

As pointed in the answer above on a similar question there is 'about:profiles' internal page in Firefox, so one can manage all the profiles (add/remove/rename/etc) and then during normal start of Firefox select which one is needed to be started.