I changed the favicon for my Community. I would like to have a favicon for my internal users who are accessing Salesforce Enterprise. Is this possible?
This may not be exactly what you are looking for, but it is a Chrome browswer extension that will override the default Salesforce icons:
According to the description:
Overrides the standard salesforce favicon with one colored based on the org. Also, displays a separate icon for sandbox instances.
Automatically updates the icon for salesforce.com to display a colored cloud icon based on the current instance. Also, includes a separate icon with an 'S' to indicate sandbox instances.
Now supports Lightning Experience!
Also, it may be possible to customize the solution that the author has provided.
UPDATE: I am editing the answer in response to a comment. I have no affiliation at all with the author of the Google chrome extension I cited.
I did confirm that changing the "favicon" in any html page requires a modification to the HTML tag. The Salesforce platform is never going to allow code that manipulates their generated HTML markup.
A browser extension to customize the icon may be the only way to do this, and this browser extension is fairly highly rated. A potential downside is that it requires access to read and modify data from websites that you visit.