How frustrating. Seems like the school could solve this quite easily by asking the other girl's parents if this is her hoody or not. It's quite likely the staff person made assumptions that aren't true.
If the school isn't willing to do that I'd call her parents and in a calm interested in the truth way ask if the pink hoody in question is her daughter's. It's likely that they'll be honest.
My grandson came home with a jacket one time and later shoes that weren't his. The teacher said that he was given them from the clothes closet and that they know they do not belong to anyone else. So my daughter has accepted them as his. Something like this could've happened with the other parents. I'd ask.
Other than that, I'd let it go. Ten dollars is not worth the aggravation and frustration. I'd just be sure my daughter's clothes were marked with a permanent marker so that I'd feel more comfortable about it not happening again.
I agree that this is also not worth the school making changes which are confusing to kids and teachers as well as involved parents.
And.....I would present this to my child as a mix up not only in clothes but in communication instead of as a theft. You're right this is a learning experience for her. It actually does not sound like a theft to me. Also saying it's a theft has a connotation of your daughter being a victim and I wouldn't want to encourage that feeling. One can sometimes do something about a mix up, tho not always. There is little that can be done about an out and out theft.
I suggest that talking with the other mother has the potential of clearing this up.