Being an American-born killer doesn't preclude one from getting a social security card or driving license in the the US either. Nothing strange about that either, especially since SSNs and DLs are not by themselves proof of much of anything other than applying for a SSN and qualifying for a DL.

There can be difficulty in flying within the US border for illegal aliens, although generally it should not pose much of a problem if familiar with how things work in the US and at US airports. Of course most illegal aliens are not fully aware of how things work (and don't work) in the US and at US airports since they are still in the learning process -- being rather new to the country for the most part.