What significance does the James Bond theme have in the movies?

While I cannot provide any hard proof or reference to back my claim, I would very clearly say that this was nothing else than a little self-refential side joke to lighten up the situation and which doesn't have much more significance than to make the audience smile at the use of this famous theme inside the movie and the little break of the fourth wall it implies.

You also have to see it in the context of the rest of the movie. Especially the Bond movies with Roger Moore are filled with many comedic elements, which also sometimes are of a little self-mocking nature, and Octopussy is a very good example of that trend.

So I don't think there is anything else behind that scene than a joke whose actual plot-relevance you're not supposed to reason about. It seems very unlikely that James Bond or the MI6 have an actual musical theme in the universe of the movie that they are aware of. James Bond just doesn't watch James Bond movies.

The theme is a self referential joke, but in the example also a sign post. The contact Bond meets is Vijay Amritraj a famous Indian tennis player in a cameo.

The use of the bond theme is to draw attention to him, and later when he is driving Bond and they are attacked, he uses a tennis racquet to defend himself.