FEZ

As the player, do you interact with a character or an avatar and how does this affect your involvement in the narrative?

The protagonist in Fez is introduced with their own name, environment and goals. There is no character creation and these facts remain unchanged regardless of who is playing the game and they are what defines Gomez. But Gomez is also the extension that allows the player to interact with the game, an anchor that has its own attributes, but also embodies the player's will and way of playing.
The narrative itself is quite  linear despite the three dimensional space mechanic, and told through very simple dialogues with minimum interaction between Gomez, the geeser, and Dot; but it is understood that the player, through Gomez, is given the mission to save the world thus resulting in a simple common goal.
There is no way that the player can progress without going through these dialogues. But they may choose to ignore and pass the text, and they would still be able to understand that they need to gather cubes to make progress.
As shown in the first minutes of games when a mysterious and unreadable font is seemingly explaining what could possibly be the background story behind the game, it seems that Fez does not need to explicitly give its narrative. In a similar way, it is unnecessary for the player to talk to other NPCs in the game, as doing so does not usually result in any action, choice, or vital information.
The game relies more on the player being intrigued by the challenging complexity of the puzzles and drawn by the beautiful imagery rather than their interest in the narrative itself. The environment is the main focus in Fez and is treated almost like a character: after getting to know all the faces of each environment level and the possibilities it offers, the player understands how to solve the puzzle.