Why does the app call my FireTV devices by wrong name?

When you start a taped show, the iOS app gives you the choice of playing it on another device on the home network instead of the iOS device.

But it doesn’t have the correct name for some of my FireTV sticks.
Instead of the new name I’ve given to the FireTV Sticks - i.e., FireTV in Livingroom, FireTV in Bedroom, it shows up as “Lee’s FireTV Stick”, which is just the default name Amazon gave it when it shipped it.

How do you change the name of the streaming device that the iOS app recognizes?

Try restarting the Channels app on the sticks after you change their name.