You say, "Who is Person A?"

The expression "Person A" acts as a proper noun, the name of a person. "the" is not used in such cases.

Giacco: Hello, Bill. How are you?

I say, "Who is Giacco?", not "Who is the Giacco?"
I say, "Who is Bill?", not "Who is the Bill?"

"Giacco" and "Bill" are names by which we can identify the persons in question.

Similarly,

Person A: Hello, Bill. How are you?

I say, "Who is Person A?", not "Who is the Person A?"

"A" is an abbreviated way of saying "Person A", so the same applies to "A":

I say, "Who is A?" or "Who is Person A?", not "Who is the A?", or "Who is the Person A?"

I hope that helped.

CJ