Note 1: (for nobody would know who I was here) Why is it important that no one know who he is, if he wishes to dance with Fanny?
The reason is that dancing was generally considered to be one of the stages in the courtship between a man and a woman, with the eventual goal being marriage. A ball was one of the few occasions when a man and a woman could flirt with one another and carry on a private conversation--a couple would dance together for perhaps half an hour at time, giving them a fairly good idea of one another.
Because there was no chance that a brother and sister would marry, it was considered improper for them to be engaging in an activity which has the end-goal of marrying your partner.