The repeated use of this annoying speech habit is part of Gatsby's affectations. Nick is annoyed by the phrase and we can see this habit of speech as part of what makes Nick distrusting of Gatsby initially. Nick does not trust Gatsby, in part, because he does not like him and he does not like him, in part, because Gatsby constantly calls him "old sport".
Gatsby's use of this phrase demonstrates a selfish or myopic demeanor in him. Gatsby is not genuine with Nick and so uses a generic and generically "chummy" term to express his mild affections for his neighbor.
Also, we can see that Gatsby wants Nick to like him. In the baldness of his intentions and the poor way in which they are carried out, we see that Gatsby is not socially adept, even if he is glamorous, powerful, and successful.
The repetition of this term from Gatsby serves to emphasize the importance of Gatsby's appearance to others, besides Nick. His unoriginal use of language feeds into the perception that his is inauthentic.
He calls him "Old Sport" because Gatbsy wants to live the "old money" lifestyle and have the characteristics of someone with money. Over his time at Oxford, which we find out is because of his army career, he has probably picked it up from people who he was surrounded by. The importance for Gatsby to impress Daisy is massive, therefore he has to imply that his lifestyle is "Old" - "Old Money" and "Old Sport"
he never says this in the book
cuz hes gaeeeeeeeeeeeeeeeeeeeeeeey
hes actually not gay but more lesbian