I got a clone of a nice oregano from someone recently and this oregano had apparently managed to survive through a frosty winter outdoors in the south of Ireland. I was very impressed as I've always had difficulty with oregano so I'm wondering if this character would be maintaned in the clone ?
Think of hardening and hardiness as two different things. Hardening is a complicated process involving thickening of the cuticle, production of protective pigments, root development... You can consider hardening to be a tropism, a response to a harsher, drier environment.
You can think of cold hardiness as the plants ability to not freeze, which is greater in established plants, and plants that have a genetic predisposition to not freeze.
So as an answer, you will inherit the cold hardiness, and some of the hardening that include changes in the existing growth... But maybe not the roots, depending on how it was prepared.