I have never heard that, if a US citizen is driving. I can't recall ever being asked for a registration except at the Aduana stops and they just want to confirm that you own the vehicle, not that it has a current registration.
The Federales have been creeping around town in recent months stopping locals who are driving with US plates or no plates. They just extort $20 or so and let the people go. A Mexican is not permitted to drive a vehicle with US plates, unless the owner of the vehicle is in the vehicle at the time or if they are married to the owner. If you let a Mexican take a vehicle and they are stopped the vehicle will probably be impounded. It's not even that simple though. An Abogado friend who works for the Hacienda (IRS) told me that Ana could drive the Titan across Fremont then down to the first cross street to Josefa with no problem but if she turned on Fremont to Josefa she would be in violation, the vehicle would be confiscated and I would have to pay a large fine. ??!?!?!!?
If they buy a US vehicle they are supposed to go through a very expensive process to import it, most don't bother due to the cost. The cost often far exceeds the value the vehicle. There are LOTS of US tagged vehicles with expired registrations driving around here.