AFAIK there is no way to obtain the apple id being used, as it would work as an excellent cross-device persistent identifier. However many people would just use their email address which you could get asking them to register, connecting with facebook and other legitimate use cases.

I thought that too, but further in the article, it explains some official dialogs don't have the AppleID.

One way to get the email:

At the start of the phishing app, ask the user to enter in their AppleID, then store it.

Sometime later, present the user asking for their AppleID password.