Ugh, I know what you mean. My group juuuust passed that stage and it's wonderful...although it still rears it's ugly head from time to time, mostly from one child who likes to purposefully start fights.
Developmentally, the reason they are fighting about *their own* coats and their own daddies, etc, is because they just simply
can not understand that the other child is talking about his own coat or his own daddy--"my daddy" refers to ONE person in their little minds and they do not have the perspective taking in order to understand that other people have coats and daddies too.
That doesn't make it less frustrating though
but I always find that understand what exactly is going on in their heads from a developmental perspective makes it easier to take a step back and a deep breath and deal with it constructively.
In the situations you described, I would cheerfully leap into the conversation as soon as the second child said "My coat" and announce happily, "Look! Jimmy has his blue coat and Sally has her purple coat and Fred has his red firetruck coat and I have my black coat! We all have a coat!" and the daddy thing, when the child who's daddy did not show up started to argue about it, I would jump in with, "Yes, Josh's daddy is here to take him home. Your daddy is at work and you'll see him later! Let's say bye to Josh!"