Being an Amazon customer does not mean you cannot enjoy Chromecast on Amazon Fire tablets!
Amazon Fire tablet (including Fire HD, Fire HDX and Kindle Fire HDX) offers an excellent option of value-for-money tablet. However, Amazon customized Android OS to such an extent that it looks different from many other Android tablets.
Although Amazon proudly named the OS as Fire OS, it is essentially still Android: most Android Apps can run on Amazon Fire tablet. You can sideload many Google Play store apps on the tablet. Please follow this guide if you are not sure how to get Google Play store apps to your Fire HD, Fire HDX or Kindle Fire HDX.
If you are a Chromecast owner with one of these Amazon Fire tablets, you can also enjoy Chromecast on your Fire tablet.