In Trek, it appears as much a title as a name (a dual purpose for the term that exists in the real world). Cumberbatch could easily be using "Khan" the way Montalban did in Space Seed (if I recall the dialogue correctly enough from Space Seed to paraphrase, Kirk presses Khan with something like "Khan, just that, nothing else?" to which Khan replies something like "just Khan").
Not quite. In "Space Seed," Khan actually said "Khan is my name" to Kirk and "My name is Khan" to Marla. (Kirk also said "Name: Khan" in the later briefing.)