Can apps identify me after I reinstall an app, such as by my Apple ID?

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?

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.

iOS App store apps (on stock, non-jailbroken, iOS 10 devices) can no longer directly get the device's MAC address. The current iOS 10 APIs return a generic fake MAC address. There is no API for an app to get your Apple ID (unless you get the user to voluntarily manually type it into the app) or Unique ID in iOS 10 (only very early iOS versions, e.g. iOS 3, etc., may have allowed this).

However, some apps can store persistent data in your device's Keychain (used by iOS to store passwords, etc.) If you delete an app, it may be possible for a re-installed app to see if previous keychain contents exist. The only way to completely delete all Keychain data might be to Erase all Contents and Settings, and set up the device as new.