The parts you've circled for Chrome and Firefox are not favicons, they are site information. The favicons for Chrome and Firefox are shown in the tab right at the top, with the title text next to them.
Favicons are decided upon by the site, not the browser. For instance, the favicon on this page can be viewed in the page source:
<link rel="shortcut icon" href="//cdn.sstatic.net/stackoverflow/img/favicon.ico">