There are a couple ways of doing that, both devices must be rooted.

Method 1

Grab the APK from /data/app by using adb pull <filename>. Then you can move them to your internal storage on the Kindle using adb push <filename>, and install it like a non-Amazon Appstore app.
Note that by using this method, you do not receive updates when the app is updated (similar to a pirated version).

Method 2

You can install the Play Store onto your Kindle. This link has some instructions:

1. Grab copies of the Google Services Framework, Google Login Service, and Google Play Services and install them in the order listed. Reboot your device.

2. Download and install Vending.apk. First copy it to /system/app, then click on it to install it. Reboot your device.

3. Open the Play Store and add your Google account information.

4. Open the Play Store and disable auto-updates for all apps that were purchased from the Amazon Appstore.

5. (optional) Get Titanium Backup and make sure that the above apps are all system apps (this will reduce how often you are asked if you want to install an app from the Play Store).

6. Install Minecraft and check that auto-updates are enabled.

Sure it is possible, but how much effort it's going to take depends on how you purchased Minecraft PE:

• Google Play Store. This isn't available by default on the Kindle Fire. However you can sideload the Google applications (although it sounds like later updates or devices may not be able to connect to the Google Play Store without rooting your Kindle), or even sideloading the apps directly to your Kindle. Note if you follow the second tutorial, you technically don't need to install Bluestacks as you can backup Minecraft directly from your son's Android phone.

• Amazon app store. Minecraft is available through the Amazon app store. If you purchased it here, you can install it on your Kindle no problem. Installing the Amazon app store onto your son's Android phone would allow him to also install Minecraft with your account. Easy hassle-free installs.

There is a lot easier way to transfer minecraft to different devices. Just use the google account you originally downloaded it with and sign into that on each device. Go to google play and go to my apps. It should be there to install