A single "server" (one physical computing machine) could have hundreds of "servers" (instances of the server-side of the game software) running on it. One physical tower PC could have "10 Quake Servers", "2 HL2-DM servers", "a Team Fortress 2 server", "a Unreal Tournament server" and such on it.
Discord originally seemed to be using that era of lan-party style language for it's naming conventions. A Discord "server" is just an instanced section of the server-side of the Discord software, and not specifically related to any real-world physical machine.