The MobileNetTest app doesn't send any information or cookie about you. Technical unavoidable is the transmission of the device IP address to the test server. You can change the test server in the settings. The server may log the IP address and get some location information through it. The server may also infer some distance out of observed round trip times.
The app displays information about observed mobile network cells. To do so it needs coarse / fine grained location permission. This is because out of the cell information it is possible to infer the current location. If you don't give the permission the app can't display the cell infos. The data isn't used otherwise.
The app use two libraries to display graphs and gauges. See the about menu items for more information. Those libraries shouldn't transfer any information out of the app, but this can't be guaranteed.