On the other hand, imagine one day you try to log in to your Twitter/Facebook/whatever-the next-big-thing-is and you can't, because the company has deleted all your data upon your request. You didn't make that request though. Someone else did it, claiming to be you.
It gets even worse when you realize that people can request all the data the company has collected of themselves. What happens when somebody impersonates you and requests all of your data?
You need to have some kind of verification method that leads back to a real identity. Otherwise this can be massively abused. I doubt that even asking for a real ID is enough.