Since it was the shepherds who were arguing and not Avraham and Lot themselves, why did Avraham add that there should not be a dispute "between me and you"?
Since when were Avraham and Lot brothers?
Rashi #1: Avraham meant that they were blood-relatives.
Consequently, when Lot would go with his shepherds to graze in other people's fields, they would mistake Lot for Avraham.
Chochmah u'Musar (Vol. 2, p. 371): Why does the Midrash teach this? It teaches the greatness of Lot; also refer to 19:3:2:1**. We find that also Yitzchak and Avraham looked alike; as did Yosef and Yaakov, refer to 37:2:1:2. Likewise, the verse below (13:11) says that "one separated from his brother" (and not from "the other") to teach Lot's greatness.