Yes, the possibility of being charged with an unregistered gun only applies in those states that require registration, of the gun or the owner.
Several states require a FOID (Firearm Owner Identification Document) of some type. Others require a permit for possession of a handgun, and this can involve some form of registration.
In NY state, for one example, a permit is required just to posses a handgun, and if the gun isn't listed on your permit, you aren't legal to possess it. If there was no other crime involved, you would be charged with possessing an unregistered pistol (illegal possession) which is really only a "paper crime" however NY takes it pretty seriously.
A felon in possession is also illegal possession, but it is a different class of crime, and a more serious matter.
I often hear on the news "perpetrator charged with possessing an unlicensed/unregistered firearm". Is there a Federal registration I have missed somewhere along the way?
As far as I know there is no Federal registration (beyond the 4473 form, and what ever background check information the Feds retain, if any). However, the news will almost always report "unlicensed/unregistered firearm" simply because it is true.
True where registration/licensing is required, and ALSO true where it is NOT required. Where not required, ALL firearms are unlicensed/unregistered, its just the news doesn't bother to mention that small detail.
It may seem like a Catch-22, but its not. it is actually a fair and even application of 5th Amendment rights. Simply put, the 5th amendment protects us all from being charged with failing to self incriminate. The state recognizes this, and so doesn't bring charges they know will be dismissed.