It is actually quite simple to do once you know how to do it. However, this feature is not fairly obvious and is almost like a hidden feature. Below are the steps to how to change your instagram story background color:
Tap on the add story on the left corner of your Instagram app
Take/upload a photo
Tap on the pen/scribble option
Choose the first pen option
Choose a color on the bottom of your screen
Hold your finger on the screen
The whole screen will change color