Unix and Linux are both mainly used on servers, while Linux is far more prominent on the desktop than Unix, its desktop market share is still abysmal compared to its server warke share.
Anyway, Unix these days isn't any one system, it's more of a certification... Linux is what's considered "Unix-like", it's not true Unix but majority of things that apply to Linux apply to Unix as well and vice versa (since they as operating systems share many of the same goals and ideas). But if he was talking about Unix, he most likely was referring to Unix, though someone referring to himself as "Unix admin" and managing Linux systems as well and the other way around isn't unheard of.