In the US, at least, you're required to keep your car's current registration card in the vehicle; that has your address on it.
There is no US federal government law that says this. You might have a state law that says this, but it doesn't really mean it.
What it means is that the registration has to be in the car while it is being driven on a public road. I keep my vehicle registration in my wallet, and my wife has a copy in her purse. Since we are never without those when driving, our cars are legal.
And as for TFA and reducing what's in your wallet, I hope the writer never has to do anything involving health insurance, as without that card, you are generally out of luck.