I could quickly try it to see if the VZ app would work on my hardware, but Verizon's choice of distribution methods is preventing it.
Why does Verizon link us to the Android Market? Why can't we download the APK file directly from Verizon's website? Who owns the app, Verizon, or Google? Is VZ's contract with Google prohibiting them from distributing the app directly, or offering it to the Amazon store for easy access by Kindle Fire users?
I know for a fact that the Apple version of this app does not require a phone number. My wife has an iPod Touch, and the version of VZ Mobile Remote she got through the Apple Store does indeed allow you register the iPod device with the STB widget using a PIN number. It was confusing at first, since the app did want to send the PIN via text message (thinking we had an iPhone), but eventually we found the menu that let the TV display the code we needed to enter into the handheld device.
I believe it's Google that is requiring a phone number associated with their user accounts. It limits access to the app, but may not actually be a requirement of the app itself. Again, I could try it myself, but Goggle won't let me load it.
Google and Amazon are competitors, and not allowing tablet users into the Google store may be their way of keeping out Kindle users. All Kindle Fire traffic is passed through Amazon's servers, and until there was a hugely negative reaction from customers, they filtered IP addresses to keep Kindle users from accessing Google's apps and e-books. Tablet users are caught in a crossfire between competitive providers.
It is not necessary to root the Kindle Fire to side load apps. You can change a setting do allow installation of apps from unknown sources. Rooting is only required if the app itself wants to add files to the system folders. I have loaded apk files from the internet into my un-rooted Kindle Fire, including apps that let me use WiFi instead of a USB to load additional apps.