Well not quite as there is a lot more to an engine that the pistons and rods. You certainly couldn't advertise it as having a new engine.
Hopefully it has resolved your oil consumption issues, when they go this far it usually does.
From what I read, they seem to deal with this problem (heat damage to pistons/rings?) in two possible ways. 1) New pistons, piston rings and I assume associated bearings, nuts, bolts and gaskets or 2) New engine. I assume they take a view when they strip the engine down and if there is no collateral damage/wear they renew just the pistons under option 1).
Renewing the pistons is not technically a new engine but I suppose from a practical perspective, on a relatively new car, it is virtually as good as.
From what I've read, owners who had had these fixes do not suffer a reoccurrence of the problem.
When they first launched generation 1 of the EA888 1.8/2.0 petrol engine in question around 2007, Audi were very pleased with themselves for making the engine lighter as well as more powerful and economical. I would guess that this quest, plus having components made to a strict price, contributed to the problem.