Does the domain root also contain an index.html file with a favicon link in the HTML header (as per the example below)?
<html>
<head>
<title>My Page Title</title>
<link rel="icon" type="image/x-icon" href="favicon.ico">
</head>
</html>
With regards to your second question:
I haven’t experimented with this to verify, but I believe that Bitwarden would use the <link> element in the index.html file for any given URL to locate the favicon to use for that URL. For example, you could set up two folders on your web server, as follows:
www.mydomain.com/site1
www.mydomain.com/site2
Each folder would then contain a favicon.ico file and an index.html file as shown above. Note that the href attribute has been set to fetch an image in the same directory as the index.html file, not in the root directory. You can change the favicon image locations, as long as you update the href attributes to point to the correct file location.
The above should allow you to display different favicons on your browser tabs. However, it may not work for Bitwarden, for the following reason:
The icons server is a cache, so it will always look for the favicon there first. If not found on the icon server, then it is supposed to go to the URL and fetch the original favicon.
The problem is that on the icons server, each favicon is associated with the domain from which it came, not with the full path (subfolders, etc.). Fortunately, however, it does use the FQDN (not just the base domain), so the solution for storing multiple custom favicons would be to set up multiple subdomains on your custom domain:
subdomain1.mydomain.com
subdomain2.mydomain.com
subdomain3.mydomain.com
Each subdomain could then store its own favicon (and a corresponding index.html file).