Is is possible to dynamically figure out the position of an app's icon on the home screen of an iPhone or iPad?
To my knowledge no you cannot natively or easily do this. I know of no open source or other libraries. The reason being that your app exists in its own world, it is not in touch per say with the rest of the device. It can get permissions to read and write data but it doesn't know of itself.
Does that make sense?
When you open a website it cannot know which tab it is in the browser. Instead it knows how it was accessed and what device (physically) is using it. It knows the user-agent, the time, the browser, etc because that is information sent to it in the request. In turn the phone on launch gives data to the app in how to handle it but not for example how many other apps are running, or where it is on the screen. It's not normally considered relevant to run time. In addition it's a security feature in preventing an app from deleting or altering other apps, as well as itself. If you have an iPhone you will notice that SIRI cannot turn off google maps navigation or any other non-apple specific app. Only apps natively comparable and private party ones (ex apples) are accessible because Apple did that intentionally. They all know of their own existence and each others. However non-native in the sense of apps that do not come preinstalled and manufactured by the company creating the device are less trustworthy, in addition there are no guarantees about how they will be run by the device, where they will be, or what other apps will be there.
It is true that an app can request for another app it may be comparable with but it is up the user to handle that information.
May I ask for curiosities sake why you are trying to do this? Are there any other workarounds?
However in terms of it being physically possible, yes. I doubt that apple allows independent developers to do this however. But an example of this occurring may be gridlock where a user can move their apps around differently on the screen. The app in this case has the ability to access app position. But I believe in this case app position is about the UI and not about nested files. apps cannot to my knowledge modify information outside of their own file. Imagine if you had an app that could edit other games scores.