The Major is a cyborg.
Which means that she is a human with extensive biomechanical parts.
As a child she was in an accident and had to replace much of her body with cybernetic prosthetics.
Having so much cybernetic equipment as part of her made it easy for her to interact with the Puppet Master.
In the end of the movie we see her body destroyed and her brain installed into a new body designed for adolescents.
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Like others have said, the continuity for Ghost in the Shell varies depending on whether you're considering the manga, the various animes or the movies.
I'll answer from the perspective of my preferred version, the 1995 movie by Mamoru Oshii .
In this movie, Motoko Kusanagi used to be human , but she has doubts about it.
A major theme of the movie is precisely this: what makes a person human?
What if we replaced almost all of the body with cybernetic parts: would we still be human?
And what if we took that final step and replaced even the brain?
What then?
This is the major existential worry that Kusanagi faces in the movie.
As evidence of both her past humanity and her worries about her present condition, consider this conversation she has with her partner Batou ( taken from the script ): KUSANAGI: That robot.
Did we seem similar to you?
BATOU:
Of course not.
K:
No.
I don't mean physically.
B:
Just what then?
K:
Well, I guess cyborgs like myself have a tendency to be paranoid about our origins.
Sometimes I suspect I'm not who I think I am.
Like maybe I died a long time ago, and somebody took my brain and stuck it in this body.
Maybe there never was a real me in the first place, and I'm completely synthetic like that thing.
B:
You've got human brain cells in that titanium shell of yours .
You're treated like other humans, so stop with the angst.
K:
But that's just it.
That's the only thing that makes me feel human: The way I'm treated.
I mean, who knows what's inside our heads.
Have you ever seen your own brain?
B: It sounds to me like you're doubting your own ghost.
K: What
if a cyber-brain could possibly generate its own ghost, create a soul all by itself?
And if it did, just what would be the importance of being human then?
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The original movies, Stand Alone Complex, and the new ARISE series are all separate continuities.
Events in one are not necessarily applicable to the others.
In SAC, she has been a full body cyborg (everything except her brain is totally robotic) since her body was severely damaged in the plane crash that killed her parents when she was six years old.
She was one of only two survivors, with the other being
Hideo Kuze
In ARISE, her brain was put into a prosthetic body before she was born.
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Kusanagi was and still is human.
Well, human at their core.
Kusanagi is a full-body cyborg.
Essentially Kusanagi's totally normal human brain is encased in a cybernetic shell, and the entire rest of Kusanagi's body is mechanical.
The technology of Ghost in the Shell has gotten advanced to the point where people can directly interface their brains with computers, which is the source of many issues in the various Ghost in the Shell stories.
But what it boils down to is that Kusanagi is still human, they just make heavy use of the cyborg technologies available to them.
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First off, I have not read the manga, and I will be making sweeping statements that may not apply to the manga.
That said - Lets go through the evidence.
Technology
Human-like AI is both possible, and demonstrated in every incarnation of GiTS, though even the most sophisticated examples like proto tend to lack cyber-brains.
'Ghost-dubbing' or copying a cyber-brain is also possible (and demonstrated), but the copy is inferior.
A cyber-brain would also be pretty heavily augmented to be able to interface seamlessly with both the cyber-bodies and external machines.
You also see them outside of the body from time to time in their metal cases.
The arise series also introduces a couple of characters that have used 'AI chips' to replace brain functions in their ailing brains.
The major Origin story
The major's origin story varies from continuity to continuity (arise/stand alone complex/the manga and the movies all exist separate from one another, in spite of their similarities)
In arise, the major has been a cyborg from birth (or at least a kind of brain in a jar).
In Stand alone complex a plane crash left her in a coma; forcing(?)
the use of a cyber-body to revive her.
The movies do not make any specific mention of her origins to my knowledge.
Age Old enough to participate in the last big war...
The wiki article on the major suggests 31 but i'm not buying it.
Body Military-issue cyber-body with cyber-brain.
Capabilities One of the best hackers in the world, and having control over her cyberbody above and beyond most.
Conclusion
The major probably isn't an AI judging solely from the fact that she predates the indistinguishably-human variants AND has a cyber-brain.
It is however quite possible that the major isn't the first... heading section nine is rather dangerous after all.
The technology does exist to make copies as well as offset the ailing abilities of poor copies with AI chips, perhaps even pushing her abilities past her otherwise (relatively) normal contemplates.
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Motoko is supposed to represent the closest mankind has ever gotten to what is known as the machine singularity.
When mankind is fully integrated with machine.
In computer terms, a ghost in the shell is when a computer program begins to write it's own code, nullifying the need for any programmer.
This was believed by scientists as the first signs of a computer becoming self aware or artificial intelligence.
Only a few times have it's been seen when code has appeared out of no where.
It was always thought it was programmed by a ghost.
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After watching Ghost in the Shell movie and Ghost in the Shell: Stand Alone Complex, I couldn't tell Major Motoko Kusanagi was an AI or she used to be a human.  Was Major Motoko Kusanagi a human in a past life, or was she an AI from the beginning?
Was Major Motoko Kusanagi a human in a past life, or was she an AI from the beginning?
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<ghost-in-the-shell>
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In SAC, she has been a full body cyborg (everything except her brain is totally robotic) since her body was severely damaged in the plane crash that killed her parents when she was six years old.  However, In ARISE, her brain was put into a prosthetic body before she was born. She probably isn't an AI because she predates the indistinguishably-human variants AND has a cyber-brain.  Essentially, Kusanagi's human brain is encased in a cybernetic shell, and the rest of her body is mechanical.   Kusanagi is a human cyborg with a mechanical body and a cyber brain. However, she is not an AI, though she closely represents one.
Kusanagi was born and still is a cyborg — but she is still human. Given she has a cyber brain, she is not an AI, though she comes close to representing one. A plane crash left her body severely damaged meaning her body is now mechanical.