FWIW in Chrome I can use this to locally override the default favicon:
chrome.google.com
Maybe other browsers have similar options or extensions?
Edit: Firefox seems to have something similar:
addons.mozilla.org
To be honest, I don't really pay much heed to the tab icons myself. Maybe others do?
Favicon Changer - Chrome Web Store
Lets you change favicons for your bookmarks, single webpages and entire websites.

Edit: Firefox seems to have something similar:

Favicon Switcher â Get this Extension for ð¦ Firefox (en-US)
Download Favicon Switcher for Firefox. Replace ugly favicon by using your own images! Favicon Switcher allows you to customize the favicons for the sites that you visit. All you need to do is simply giving a url pattern and picking an image. Happy Browsing!