Yes. A big one.
Part of the reason why I took an interest in Rei Ayanami (the character in my avatar) was because of some personality similarities that were almost frightening to observe the first time I watched Neon Genesis Evangelion.
First, Rei's sense of self worth is extremely low, which leads her to engaging in behavior that could be classed as suicidal or "death-seeking" in its nature. Her basic reasons for this (her origins) may differ from mine (past events), but the appearance is basically the same. I would not consider myself suicidal (i'm too stubborn to ever try to kill myself), but a fascination with "a return to nothing" does indeed exist in my psyche. So does a near-suicidal disregard for life that leads me down the road of self-destruction in order to accomplish objectives, much how Rei acts in several NGE episodes.
A second similarity is having an extremely introverted personality. It is not easily apparent online on this website because I post too much, but it is very much the reality in the real world. I explain this major discrepancy by my belief that online and offline communication are very different things, and that it is much easier for me to express thoughts in writing than in conversation. Like Rei, people tend to say that I am "hard to approach" or "in their own world", and yet there is something that is strangely compelling about Rei/myself that keeps outsiders interested. I tend to say little, but I am always in a state of deep thought about many things, often about personal soul-searching questions that few others ever learn about.
The third similarity is, and perhaps the most disturbing, is a difficulty in being able to express emotion, or at least to feel it genuinely. Most critics of Rei as a character point to her seeming "dead" or "emotionless" nature. The major reason I see this differently is because I sort of know how it is to live in that way. I can get "angry" or "happy" in a way, but I don't *really* feel those emotions to any degree. I could fail an exam or pass it with a perfect score and still feel the same thing inside: little to nothing. Emotion may come across in some of my writings here, but chances are I am not really feeling it, especially when regarding anger. I can "think" I should feel something, but physically there is little going on. Granted, I don't think my case is quite as bad as Rei's is in this regard, but it is close enough.
In these aspects of personality, for better or worse, I am very similar to Rei, though I would consider my case to be much less extreme. Does this disturb me? A little bit, but it is just how I am. I noticed these things about myself long before I came across Rei and NGE, but it was when I saw that character that I decided to use Rei as my main forum avatar. I think its a good fit that has personal meaning to me, even if that meaning is probably a bit more uncomfortable than I would prefer.