Does my phone have to be on the same network as the TankMate sensor?

Definitely not, the sensor sends readings through the internet so that you can be almost anywhere on the planet and know what's happening. 

To do that, the sensor needs to connect to our secure server through the internet, either via WiFi or through the cellular network.

And then our secure server sends the readings down to the TankMate App on your phone.

So as long as the sensor has internet access, and your phone has internet access, then you'll get the readings. The sensor doesn't need to be on the same network as your phone, they connect completely independently of each other.

And there is no direct connection between the TankMate sensor and the App on your phone, other than when initially setting up the WiFi model.