Prior to setting a name in Siri, it does not call you by a name. For example ("I'm not sure what you said" rather than "I'm not sure what you said, insert name here"). So how do I revert to the behavior of not getting called a name after I have set one. There is no option to do that in Settings > General > Siri except changing to a different contact.
It's easy to make Siri forget your name by creating a sacrificial contact:
Open Contacts and tap the plus sign to create a new entry. Enter any text you like for the name. Tap 'Done'.
Open the Settings app and go to Siri's options under the 'General' tab. Tap on "My Info" and change the contact from you to the one you just created.
Go back to the Contacts app. Select the new contact, tap on 'Edit', scroll down until you see 'Delete Contact', and tap on it.
That's it—you're done. Siri's settings will now show "None" next to "My Info".