apple-idiosipad
I'm trying to make this app "forget" me and not be able to identify me when I install it again. I'm on my iPad Air 2 with iOS 10.1.1.
I read on another question that they cannot see your unique device identifier anymore. Are they able to persist data in any way (I have iCloud turned off), or can they view my Apple ID? Do I need to perform a full reset?
No, you need to wipe the entire device using the settings found under Settings > General > Reset > Erase All Contents and Settings. Simply deleting items from iOS applications does not provide any kind of secure deletion, and those files can easily be recovered (although that will be harder and harder as those blocks get overwritten in time).
This provides a one-pass secure delete call that can be liked to OS X's "Secure Empty Trash." While most 3rd party software won't be able to recover your data, government units have far more sophisticated tools that can retrieve data from far worse cases (even physically damaged storage).
This feature was introduced in iOS 2.0 and you can read more about it over at AppleInsider.
Try setting the iPad in recovery mode:
connect the iPad to your computer
turn it completely off
press the power button (top button) for three (3) seconds
without letting go of the power button, press the Home button
keep them pressed for ten (10) seconds
after the ten seconds, let go of the power button without letting go of the Home button
keep pressing the Home button until a message pops in iTunes saying
iTunes has detected a device in recovery mode, you must restore it before you can use it.
restore the iPad
Try doing this a few times if you dont succeed the first time.
Best Answer
Since you haven't identified the app in question we can only provide fairly generic advice. One thing apps can do is view identifiers such as the media access control (MAC) address. This is not to be confused with Mac as in Apple Mac.
The MAC address of a device is a unique identifier assigned to network interfaces for the purpose of communications. Most IEEE 802 network technologies, including Ethernet and Wi-Fi, use MAC addresses. Devices that have MAC addresses are not just computers, smartphones and tablets. They can include anything that connects to a network (e.g. printers, smart TVs, weather stations, etc).
If you're concerned about a particular app, you should take the time to read its privacy policy.
If your question is really about something else, such as wanting to change accounts used for the app, you may be best to contact the developer directly or use their support pages.
[EDIT]
As suggested by bmike in comments, I'm updating this answer to include info on changes that Apple are enforcing as of the soon-to-be-released iOS 10.3 update.
Up until now, keychain data in iOS remained on the device even after the app was deleted from the device. After upgrading to iOS 10.3 deleting an app will also delete all associated keychain contents for that app unless its keychain item is shared with other apps. In that event, the keychain data won't be deleted until those other apps are also deleted.
Apple's intention with this is to further protect user privacy by ensuring that any information that can identify a user is not left on the device after the app that saved it is removed from the device.
In a nutshell - anyone concerned about their personal data should upgrade to iOS 10.3 if their device supports it.