My neighbor has a 10 year old Italian unknown fig tree and the way
he covers it, is with a tarp right over it, with no kind of instillation inside.
Is that a method of covering a fig tree as well? Or is he just lucky
that his tree is very cold hardy and doesn't even need covering?

I'd say it wouldn't matter and its probably cold hardy enough without protection. I can't imagine a tarp alone would keep it safe otherwise.

Maybe the tarp would help protect from the severe winter winds that might do some damage...on the other hand I understand that a tree of ten years of age no longer needs winter protection...I have a 7yr. old tree that is getting very tiresome to wrap each year...but I'm too chicken to leave it unprotected yet...maybe on its 10th. birthday I will let it fend for itself.

I have several unprotected larger trees and the issue for me is loss of the tips of the branches for winters with long periods of below 20 degrees.  A tarp with a frame that holds in a blanket of still air around the entire tree and down to the ground does help to minimize freeze damage, but needs to removed for those periods of bliss of odd summer like days in late winter with mid 60s and above, because the sun can overheat the tree inside the plastic.  I have found that military surplus square canvas road stop tents work great. the cloth holds in the air, adds insulation and a wind break. Your neighbors will find a new reason to despise you, but the older ones are still made in the USA!

I have wrapped a tree for three years, first in a light blanket, then a tarp and so far minimal damage.