My iPhone 13 pro does not show “Action Mode” as an option

As the previous user mentioned in their post above, the iPhone 13 Pro does not support action mode. Though it is a software-based feature, it’s only available on iPhones 14 and 14 Pro. You can confirm in this article:

About the Camera features on your iPhone - Apple Support

The iPhone 13 Pro - as well as iPhone 13 - has excellent stabilization with sensor-shift OIS on the main camera, but you have to upgrade to the latest iPhone series to get even more stabilization with action mode.

Seems Action Mode was newly introduced with iPhone 14.

Eg: The "Related Article" above your question says:

"With Action mode on iPhone 14 and iPhone 14 Pro models ... "