They became friends in the movie, too. And no, they don't make it look like Dan White killed Milk because he hated gays. Did you watch the film? They make it very clear that White felt betrayed by Milk for his lack of political support after they became friends, and that White eventually became unstable. I guess Diane Feinstein isn't very good at interpreting movies.
Besides, they don't hand out the Best Actor Oscar for historical accuracy. Do small inaccuracies cheapen the acting job he did?