It doesn't really matter if you can hear the clutches working like a previous poster stated. That doesn't define anything. The DCT requires not one more ounce of driver involvement than any other "automatic car with paddle shift" out there. Compared to an automatic 328i, there is the same involvement to shift from a driver's standpoint, which is directing the shifter in a certain direction. So from a driver's standpoint, the DCT is essentially an automatic.
On the other hand, from a technical perspective, people could still consider it a manual IMO. B/c yes it lacks a torque converter and gains all sorts of fun little gizmos they write about in the manual but no one ever knows what/where they are.
So in conclusion, there is no more connection to the car with the driver than any other auto on the market. So basically it's an automatic with a fancy name.