The version in the Amazon Appstore is outdated. But I think (not 100% sure) it should be possible to either install Google Play Store and download Tiny Tower from there or go to an APK site like this one and download a current version of the game.
If you are on android you can use an app like Parallel Space to run multiple copies of TT and have separate TT accounts. Otherwise you can use separate devices such as a second phone or tablet to control a second account. You'd mainly use it to house your gold bits when you rebuild.
Isn't the OS the Amazon devices use just a modified Android version? I think it may be possible to install the Google Play Store and download the app from there or use an APK file like this: https://www.apkmirror.com/apk/nimblebit-llc/tiny-tower/tiny-tower-3-6-6-release/
This is the auto clicker I'm using:
https://play.google.com/store/apps/details?id=com.truedevelopersstudio.automatictap.autoclicker
❗Important: you have to make sure that it stops after 1 cycle. I suggest delay of 100 milliseconds until the next tap.
(I accidentally placed number 3 on the wrong position first, but moved it to the correct position as you see in the video)
It's pixel art. I recommend you either this https://play.google.com/store/apps/details?id=com.studios.pixelstation or this https://play.google.com/store/apps/details?id=com.PixelStudio if you want to draw yourself.