They are "technically" (ie mechanically) a manual transmission. Correct. They are sold as automatics though, and most everyone refers to them as automatics, however, most magazines call them "automated manuals", "semi-auto", or just the trademarked "DSG", "PDK", etc.
Last time I tried to explain my transmission to someone they did this:. Now when someone asks if it's an auto I just shrug and say "sorta, ya".
Lol, exactly. Not too many people understand the mechanical differences between manuals and autos. So if a gearbox can change on its own, they call it auto right away even though the mechanisms involved have nothing to do with automatics (in the automotive context).
I hope to get a VW GTI mkVI soon with the DSG so I bet I'll find myself in the same situation haha.