I've heared some people say that "you are not V, since V is a separate person with his/her own personality", and the player is not completely free in defining who V is. So me saying "You ARE V" - at least understood that way - might be a bit inaccurate: V indeed isn't you, the player.
Still, i stick to the point: You are V. Not by being the same person - but by experiencing the world of Cyberpunk in his skin, seeing it through his eyes.
This kind of direct but foreign sourced experience is actually a staple of the cyberpunk genre: Think of movies like "Strange Days" where you use a SQUID-Device to record and experience life of other people, or SimSense / Better Than Life chips in the world of Shadowrun. Or, in more conventional media: you are the actor directly improvising the role of V - not just a person watching an interactive movie guiding V.