Which is more correct?
He's talking in the third person.
He's talking in third person.
Does it matter?
Both styles are used. In this NGram, the phrasing with the is more common but both are used:
It sounds natural to say either, but I would suggest using "in the third person" because it is more widely used. This may be because "third person" is a narrative mode in which you write. That is, you write "in the mode of the third person" so you write "in the third person". Since NGrams pull from books, it is possible that "in the first person" is more formal because of this reasoning. That being said, a quick Google search for either phrase will bring up plenty of examples both ways, so I wouldn't worry too much about which is correct.