It's far from clear exactly what information Twitter, in its longtime effort to combat spam, has already collected on its users. Throwing a phone number into the mix expands to range of activities that can be unambiguously tied to the same individual. Given that Twitter certainly knows IP addresses of its users, it's trivial to do things like link site visits directly to an individual answering the phone number.
Unfortunately, most people simply don't understand the implications of any of this. They don't understand what an entity, "authorized" or not can do with this kind of data. They are far too trusting of governments and other powerful entities to do the right thing. They have not been paying attention to the steady erosion of civil liberties around the world and can't conceive of, for example, ever ending up in prison for some lame post they made 10 years ago.