First off: I use the server finder tool in the game, so I can't use the server browser to find my server unless there's something else I'm missing there, it's entirely possible.
The "add current server" buttons are greyed out at all times for me whether or not I'm on a server. What I want is a way to find my current server's IP via the command console. I don't need a command to hit a button to favorite or blacklist, just one to find the IP, I can put that in manually.
Best Answer
If you simply want just the IP of the server you can try this:
When playing on the server, go into your console (using the ~ button, next to "1" on your keyboard, after enabling the console in your options -> advanced). In the console, type "status". This will give you some information about the server and players on it. The server IP is in the line udp/ip.
In short: Enter console. Type "status". Check the UDP/IP line
Alternative: Click on servers -> History -> Refresh -> Sort by date -> Right click server -> Add to favorites