Favicon will have nothing to do with SSRS on its own, if it is in the root folder of the website, most browsers will automatically extract it. Another way to add an icon is to add the <link> to the <head> web page, but I do not believe that you can change the report manager website to such an extent.
Make sure you put the icon in the actual root of the entire website, and not just in the root folder of the report manager, as it usually maps to http://reportserver/reports/ , and instead places it on the actual website, which as far as possible I know by default for IIS there is c:\inetpub\wwwroot\
Update: I did not know that SSRS 2008+ is no longer required by IIS. It is interesting.
Since 2008, SSRS has been using HTTP.sys for server HTTP requests so that it does not require IIS. This means that you are very limited in how you configure it. This forum post on MSDN Social seems to imply that you cannot put anything in the root directory of a website unless you have IIS. This blog article provides much more information about http.sys and how it works.
MSDN has a page configuring the http.sys kernel-mode driver and reserving URL namespaces . But all this suggests sending the query to the SQL server anyway, so I'm not sure if this will be very useful.
You can try changing the virtual directory that SSRS will serve to the website itself and put the favicon.ico file in the \ Reporting Services \ Report Manager \ folder, but I’m not sure if this will work, and I can’t check it for this moment.
Of course, the easiest solution is to simply install IIS and use it. Then it will be easy to add favicon.ico to the right place.