Again to me that seems like the person that I "am" as other people would define me. I define myself much more simply-- I just perceive myself, that's the point about what consciousness "is". I think that is the key point you are missing here, and I'll bet that you perceive yourself that way too, independently of how others define you, or how you behave in various situations, or your DNA sequence. That does not sound like a "sense" that you describe, but rather a property of some kind.
The problem word in the phrase is "sense". In one usage, like a "sense of touch", sensing only means the ability to detect, like the "sensors" on the Starship Enterprise. Another usage, like a "sense of beauty", is a higher-order phenomonen that links what is being sensed to how something is reacting to it. A sense of identity combines the thing being sensed with the entity reacting to the sense, until they are indistinguishable. If you can distinguish the sensor from the sensee, you are not talking about an identity.