Why does Katniss think Peeta is right about her and Haymitch being alike

Because both had the personality shaped by the door of losing loved ones at the hands of the capitol.

Explanation:

Haymitch has already won the Hunger Games and was cast to mentor Katniss and Peeta in the new edition of those games. However, he and Katniss don't get along from the start, they both don't like the way they deal with the situation.

Peeta believes that the explanation for their dislike for each other is because they are very similar. Over time Katniss ends up agreeing. That's because she realizes that both had their personalities shaped by the same reason: the death of loved ones at the hands of the capitol.