At first I thought about how characters have evolved throughout the gaming history. They started out rather modest. Remember games like Super Mario Bros, Pacman, Space Invaders? Everything we put so much emphasis into nowadays, were more or less unimportant in the 80's; graphics, character design, storylines all stood back for what was really important, and then also the only thing in which people actually had the technical capacity to make a difference - gameplay. I'm not trying to be all nostalgic and say that games used to be better. Games were the way they were because of limitations, and now that those don't exist anymore we can afford to demand more than just great gameplay from games today (although I will still argue that good gameplay is the most important ingredient to a good game). The games that actually had characters and storylines, like Mario, had those elements work more as tools to allow us get to the game, rather than adding anything of their own, much like a baseball bat or football. The best way to examplify this is the evolution of rpg games. In the first Final Fantasy games you play a very anonymous bunch of people, simply named "White Mage", "Knight" and so on. They're not actually characters yet, they are a tool for you to experience the game. These would however eventually involve into becoming not only tools for you to play with, but tools with which the game designers could tell a story. So we've got a series of events here. First off you had games with basically no stories, and without stories you don't really need characters. When storylines became an important part of a game, designers choose two difference ways of telling it. Either you had games with stories, in which you were the main characters. Or game designers decided to let the players shoulder the mantle of someone else, and tell the story through that character instead. These two schools, or ways of telling a storyline still exist. Either the game designers aim to have you, the player as the main character, or they want you to become an already existing character in the game to experience the story. I'd say the latter is more prevalent, but which way is the best? It depends completely on the game of course.
Let's take a look at another very interactive way to play stories - playing with toys. You can either shoulder the role of Batman, or play with an anonymous plastic figure. Most of us have had no trouble playing with characters that already have very set personalities, in fact I think most of us preferred to be someone cool rather than someone who no one knew anything about (I had dibs on the Venom toy) . The big difference to playing with toys and playing a game is that when playing with toys you decide the story and when playing a game you rarely do. When playing with toys you also get to choose which character you want to play in your story. In games you can affect the story, but you can never change what it is about. Because of that we need to be handed a character that suits the story. For some games, that is just impossible to do, and that is the key here.
The answer is - the goal of the game. When I start playing a game like Final Fantasy, or Mass Effect, or Crysis, I know that I will take part in a couple of hours of a story. The story can have different endings, but in most cases there are only a couple of ways to experience the game. Even games that are considered "sandbox" games, like GTA, have put loads of limitations to how you can experience the game. Whatever person you happen to be, there are only so many ways to play the game. This is not true with a game like World of Warcraft. When you first start it out, there is nothing about the game that forces you to focus on raiding, on instancing, on being a healer, on spending all your time chatting with your friends or herbing or just playing the Auction House. Because WoW has no defined goal for the player, no story that has to be told, we don't want to be handed a tool for telling us something that we're not interested in. If there was an option when you started the game to choose "Playing the Storymode" then I think I'd like a set character that helped me experience the big story that exists in WoW. Imagine if you could really be some hero, instead of just nameless spawn from Elwynn Forest, when fighting through the hordes of evil? But now, since there are thousands of ways to experience WoW, it would be impossible for Blizzard to design a character that fits all those play-styles.
For any other game, however, the character is an important piece of the puzzle to tell the story and add to the overall experience (or at least I think so). Even though Crysis 2 is an overall good game, playing a nano-suit shell, void of any personality is actually a flaw of the game. Not to mention all the awkward situations that rise when people talk to you and your character doesn't answer or situations that could've been solved in a simple way if your character had just talked. There are problems with designing a ready character for the player - if you fail in making the character insteresting, the game will take a major blow too. I've quit reading many mangas just because I couldn't stand the main character, even though everything else is just fine (I usually manage to get through movies and books because they're a lot shorter). Naruto is one example. But I still think in games it's rather difficult to create a character so annoying that people would've preferred a voice-less robot. Well, I can think of one good example - Tidus in FF10. Overall I don't think that people have to agree with the character they play in everything, since the character is part of a storyline. You're not supposed to "be him", he's not supposed to represent you. He's supposed to be a part of your game experience, just as much as any other character, you just happen to see everything from this persons eyes. A well designed character can add tons to a good storyline. Just look at Cloud.
All pictures from Wikipeda.