Private tags for private roads. The state owns the public roads and represents the public. We’ve all agreed to pitch in costs and register vehicles on our roads. If you want to use them, that’s all you have to do, otherwise it’s trespassing.
Buggies are animal-powered, and by law (in nearly all states) that exempts the vehicle from needing plates. You don’t have to be Amish to “take advantage” of this “loophole.”
Around here you can drive “farm implements” on roads without licenses. I know a redneck dude who drives his tractor to go to the legion since he lost his drivers license.
Indiana actually does require a license plate and fee for a horse drawn carriage. Most other states do not at this time, though there are proposals in some.