The redundancy is emphatic just as might be, "Yes, indeed." I can see this being used in all English dialects and regional variations.

I am, I am - I think he's just emphasizing, reiterating that yes, he is partial to key lime pie. Sometimes one might repeat something for emphasis.

I don't know the specific movie, but you could also have a weird situation (maybe comedic) where the guy is schizophrenic and is answering "I am" for both his personas, and likewise wanting 2 pies to go for both his personas =).

As others have said, it's conveying that the person is saying "Yes".

But more than that, the doubling of "I am" often has deeper meaning, depending on context and intonation. In your example, it sounds like they're playing a guessing game. So "I am, I am" if spoken slowly would indicate the correctness (and perhaps a bit of surprise) of the question itself. The previous comment by that character ends with "as a matter of fact" - which means the same thing, namely "your question was correct".

If spoken as an exclamation, it would underscore the strength of the answer ("Are you hungry?" "I am, I am!") To a question in the negative, it would be an emphatic rebuttable to the question ("You're not hungry?" "I am, I am!")

It isnt just British English. I am American.

If someone says, "Are you hungry?". The full response is "Yes, I am hungry."

You can also say any of these or a combination,

"Yes."

"Yes, I am."

"I am."

"I am hungry."

They all are just the same thing. They just mean, "Yes, I am hungry.".

I think you are right. It just means "yes" in this context.

Or in other words it's the answer to the previous question. But the question and the answer are shortened. More detailed it would be:

"B: Are you partial to key lime pie?"

"A: Yes, I am partial to key lime pie."

Another example:

"B: Are you spanish?"

"A: Yes, I am."

or simply "A: I am."