Assigning a favicon is quite simple in Typo3. you can just use the following Typoscript to assign a favicon.
page.shortcutIcon = pathToFolders/favicon.ico
In case of multi-domain setup there are various ways you can achive this:
1. By adding the above code in the Typoscript Template Record of root page for each domain.
2. If same root page is used used for 2 domains we can write a conditional statement and check the domain, and change the Favicon.
[globalString = ENV:HTTP_HOST = *mydomain.com]
page.shortcutIcon = pathToFolders/favicon.ico
[global]
[globalString = ENV:HTTP_HOST = *myotherdomain.com]
page.shortcutIcon = pathToFolders/favicon2.ico[global]