I think it only can if you let it. The only power things have over you is the kind that you give. If you're not weakwilled, or if you don't want it to change you, it can't, or, in a more open way, it can't have such a strong influence as it otherwise may have.
I have very little sense of identity whether I'm making art or not, so for me no, but for other people I guess it could work both ways, their art may just be a reflection on themselves, or they may learn something about themselves through a happy accident.
It is also a fascinating tidbit of data to see that the majority of people on art sites have this view point(of confusion of terms/definitions?) and reveals a great deal about the mental conditioning or quirks that cause artists to be who they are. It is also incredibly disturbing as it might mean a complete shift (perhaps in artistic ability as well) if their illusory viewpoint was suddenly shattered by reality. It is also shocking that the phrasing of such a poll and its respondents answers reveal so much about subjective dangerous of enlightenment(and the willful anti-pursuit of it).
It depends. If the person is emotionally attached to a character, setting, plot, idea, moral etc, then most likely. People have a tendency to sub-consciously idolize other things. For example, with my writing, I can't say that Rowan or Ky from 'Angels vs Demons' changed my sense of who I am. However, I can say that some of my writing, not seen here on DeviantART has. Because I'm emotionally attached to those characters. It's the same with games and the art within them. If I become attached to a character, whether it be a villain or a hero, then I may pick up a few behavioural habits of that character. When we sub-consciously idolize what we see or become attached to we also tend to sub-consciously pick-up those behaviours. However these sub-conscious tendencies are not found in everyone and it truly varies, quite a lot, I have seen it in my friends and even people I meet online or out in the street.
Also, if we don't fully change our sense of our selves and change behaviour, we will certainly question it. All depends on what kind of life you live, what you had, what you do in your spare time. Gamers, writers, painters, enthusiast readers, inventors, drawers, photographers and any other artist/enthusiast you can think of will have a higher chance of the sub-conscious tendencies, or change in general to someone who spends less time being artistic, relaxed, gaming and more time working, stressing and worrying.
However like I said, it all depends. You could do it sub-consciously, you might not, are you emotionally attached to something about that art whatever it may be? Are you more creative, relaxed, gamer-ish, an enthusiast? Or are you the opposite? Serious, leaning more towards work? Think about it and then develop an answer. The mind is easily influenced, but hard to notice itself being influenced until pointed out.
I rather depends on the person, the art, the setting and several other factors. The one thing that really makes this question, is the definition of art... as there is (as far as I know) no true definition. For example, I think some games are art, but others are not. In both cases they have changed me in some way, some more than others. So would I say that "Art can fundamentally change your sense of who you are?" Yes, No and Maybe