The Kindle application for Mac (version 1.23.3) launches and works fine on my late 2017 27" iMac 5K running macOS 10.14.1. It does pop up a note that the application is not optimized for Mojave (because it is a 32 bit app) but works just fine.
Kindle for Mac stopped working properly when reinstalled on Mojave 10.14.5 an instead of showing the Kindle for Mac icon in the Application folder it shows what looks like a shortcut and launches the Kindle.app DMG which asked if the application is trusted before it will launch. It worked fine on Mountain Lion 10.8.5 and higher but on Mojave it is suddenly mounting the dmg and loading the dmg every time I start the computer instead of just having the Application Icon in place to click. I contacted Amazon via Chat and after pointing me to some non apple site on how to uninstall an application they said to contact Apple. I told them I'm using Kindle For Mac (Macbook) and it is AMAZON's application. I gave up. If you go to the Amazon page you have three options, to go to the Apple App Store (where you can download the iOS version for a phone), or the Kindle application for a Kindle device, or Kindle for Mac. The version is 1.26.1 (55093) Updated 7/10/2019 7:55am