There is no way they can make you register and insure a car that is not on the street. I would have insurance on it anyway. At least fire and theft. Just call your agent and put it on. It won't be but $100.00 a year max.
Oh yes they can. I live in Maryland and though not always enforced, there is some law that requires you to have your motor vehicle registered. Any unregistered vehicle is considered "junk" by the county or state. All it takes is one neighbor to complain and the next day there is a notice taped to your door giving you about 30 days to register or remove vehicles. Again, its loosely enforced and I believe is more in effect to deter people from starting personal "junk yards" in a neighborhood. I argued my case with the county rep and my neighbors and I got to know each other a little better and now all is good.
As for insurance, I use Hagerty and they let you state the value so you could keep it low. However, they may require you to send pictures of the locked garage you keep the car in.
Most of those ordinances are *city ordinances*, although there may also be some states that have similar. I had an experience 30 yrs or so ago with my town (Huntsville, AL). I had a Henry J parked on a concrete pad next to my home. I kept it locked, clean, air in all the tires, Braked so it couldn't roll. A neghbor apparently didn't like such views, and I got a notice from the city inspector. I checked in with them, found out the basis for the ordinance was 'safety'.. ie a lot of such cars could allow small children to climb inside or fall out and get hurt, was a haven for snakes, pests, etc, was an eyesore.. I explained the conditions on my car, told them it was NOT unsafe, and an *eyesore* to one person was beauty to another..
I told them I absollutely hated to see woodpiles and swing sets in my neighborhood, and that I couldn't move the car, it didn't have a motor or trans in it. I couldn't drive it, thus no need to pay for a license, and that they should go ahead and penalize me so we could hurry to court...
It never happened!