When I'm just talking about point of view in video games with my friends, I usually refer to it in relation to the control scheme. Is the camera in first-person or third-person or second-person like in those creepy Kinect games that are recording you? I don't think that would be considered second-person, but I want to think about point of view in relation to the story instead.
Video games occupy an interesting space because player actions are necessary for their stories to progress. Hyrule isn't saved if I don't collect the Triforce and defeat Ganon nor is Dubai "saved" if I don't venture deep into its heart of darkness (Spec Ops: The Line). When I read Pride and Prejudice, sure there are moments where I really feel like I am in Elizabeth's psyche, but there isn't a moment where I personally feel the desire to make clever jabs at Mr. Darcy or visit my older sister in town. These are actions that are spurred by her own motivations, Jane Austen need only make them believable. On the other hand, even though Link or Walker may have the believable motivations down pat, the player must be motivated to execute on them.
Where this sort of dilemma comes to the forefront is in games where the player must make decisions for the protagonist like in Life is Strange or in The Witcher series. Deciding how to respond or which paths to take reflects not only the character's motivations but the player's motivations as well. So I'll often reach these points where I have to choose between what I believe the character would do versus what I believe I would do versus what I believe is the right thing to do. It can get all out of sorts at times, and, at those times, the path I end up taking is almost random, solely dependent on whatever mood I happen to be in. What I'm trying to say is that while these games are about Link or about Geralt or about whoever, the player is intrinsically a part of their identities as well.
Recommended Play: Spec Ops: The Line from 2K Games