I wondered about this exact scenario with my Android auto dongle.

I never could get the upgrade to work when connected to my phone. I did get it to work connecting my PC to the dongle (which acts as a wifi access point). My PC then loaded the firmware in a desktop browser window as I was connected to Ethernet and the dongle at the same time. I may have had to set some up routes manually.

They absolutely should not get Internet access through the phone. I really hope it doesn't work that way! I searched quite a bit but could not find a definitive answer.

Edit: I read the other replies and it makes sense. The browser on your phone can make a request over wifi and over cellular, so really the browser would do the fetching over the internet, not the device.