Look into running separate sessions of your browser(s). Both Firefox and Google Chrome (or Chromium) allow you to do this, although the interfaces for doing so differ.
A simple way to do this is to use different brands of browser, e.g. Gmail in Chrome and everything else in Firefox. But... if you really prefer one browser over another, for all use, then the separate profiles thing works.
Note that in Chrome, this is now confused by the ability to change Google account log-ins. That is not a separate, browser-level profile with separate configuration.
Instead, you are looking for the command line invocation argument --user-data-dir (in *NIX, at least; IIRC the flag name may differ slightly in the Windows version).
For Firefox, there is the -p flag. IIRC, you have to combine it with another flag in order to ensure both that the profiles are running in separate invocations and that you can be prompted to choose what profile to use when you invoke Firefox.
Of course, you can create menu items / icons for these invocations to make them "clicky" and avoid having to go to the command line and enter them each time, if you prefer.
P.S. Yes, this will help you less if you insist upon clicking directly on/through links that are are e.g. mailed to you or, if you have Facebook in its own "box", posted on Facebook.
From that perspective, having per site browser extension variability might still be useful. But then, you're still looking at also controlling referer passing, cookies and other local data, etc., etc.