How does "Find My" application is working with location service changes in iOS 13
The While Using App permission means it's using your location when actively running (foreground) or in the background.

You can keep 'Find My' app in the background (do not force-quit) and it will be able to access your location.

Update: Source:

The Always Allow option, which grants an app “background” location access, has been removed from the initial location permissions prompt. 

Instead, users can select Keep Only While Using (aka “foreground” permission) or select a new option, Allow Once. 

So how does an app then get upgraded to “background” location permission? If the user continues to use the app, iOS 13 will now automatically and periodically prompt to upgrade location permissions from While Using to Always Allow.