From a quick check in TCPView, Google fiber's speed test is multi-threaded. Based on a test my with my connection (which only peaks about 45Mbps), it made 20 simultaneous connections:
Although TestMy has a multi-threaded test option (default test is a single connection), the 200MB maximum block size is not sufficient to properly test Gigabit connections as the test will complete in under 2 seconds (900Mbps = 112.5MB per second). So for measuring your peak speed, the Google fibre test will likely be more accurate.
On the other hand, the normal TestMy linear test will give an idea of what your connection is capable of with a single connection, similar to downloading a very large file with a web browser or FTP transfer. If you are getting in the 800Mbps range, your connection is fine. However, if it's much lower, e.g. below 500Mbps, then there is probably something limiting what you can achieve over a single connection, which the multi-threaded speed tests don't show, in which cause you would only be able to achieve the multi-threaded test result with a multi-threaded download manager, e.g. Firefox's DownThemAll plug-in which splits a large download into multiple segments and downloads these segments simultaneously.
You are measuring the speed from your device (pc,tablet,phone) to the test server. The ip address you are seeing is the static IP given to you by your ISP. No you cannot connect directly to their server to test your speed unless they put up a test server. The point of doing a speed test it too see if you are getting the bandwidth you are paying for from your Isp or to help you figure out local network issues.