This article explains how to Chromecast to your Fire Stick. It includes instructions for casting to Fire Stick from Android devices that support Miracast, and for using an app if your device doesn't have Miracast functionality.

To cast to a Fire Stick from an Android phone that supports Miracast, you’ll need to put your Fire Stick into display mirroring mode and then connect your phone. Once connected, your phone’s display will be mirrored on the display connected to your Fire Stick.

Here's how to cast to Fire Stick like Chromecast (steps vary depending on your smartphone):

Open Settings on your Fire Stick.