This is a good question. Gatsby used this language for three reasons. First, he wanted to give the impression that his money and wealth were not new but old. This is an important distinction, because new money is seen as less sophisticated and gaudy, whereas old money is just the opposite - refined, in good taste, and proper. So by using the language of the wealthy of England, he is seeking to give the impression that he comes from arguably older money than Nick and the others. It is important to recall that Gatsby told people that he was at Oxford. He wants people to make the connect to Oxford.
Second, the language also has an "old boy's club" feel. So, when he calls Nick "old sport," he is creating the image of an inner coterie of friendship. By using the language he is saying that he himself is part of the "club," and he is extending it to you.
Finally, all of what he does is to impress Daisy. So he is creating a persona for a girl, because of his love.
Gatsby uses this phrase in order to establish his position in society. "Old sport" is an English term of endearment, though it is not used by many. He uses this term in order to convince his peers that he is the person he is pretending to be. He wants people to believe that he attended Oxford, so he enables many different manipulation tactics, including the use of "old sport". He has failed slightly in this though, as the phrase is not actually common among the English. It is all part of the facade he presents.