Yes possession of any alcohol including moonshine is illegal.
The law states that any alcohol including moonshine that hasn't had sales tax paid on it is illegal to possess or consume and if caught with moonshine in your possession you can be fined or even serve jail time for 6 months to a year depending on the states laws.
It's not illegal to actually own a still unless you do not have a permit to operate that still.
In the state of Virginia they say it's illegal to possess or operate any distilleries or distilling apparatus without a permit.
So yes possession of moonshine is illegal since you wouldn't have paid sales tax on it and that really is why that moonshine is illegal due to the taxes.