I also think it matter what type of dress it is. If it's a strapless black number, then there will be lots and lots of people wearing dresses which are strapless and/or black. You have to expect, when you wear a black or even white dress, someone will wear very similar and at a prom, there may be lots of women in a black dress or even the same dress. Anyone old remember the ubiquitous Laura Ashley black bodice, coloured silk skirt dress that everyone had on at my university balls? Absolutely hideous on everyone, but there were literally twenty young women in that dress at the last ball I attended.
If it is a very distinct lime green with a train, then obviously if someone else has the same dress, it looks odd and like twins.
I do think price has some bearing on this, because you are worried about losing the deposit if you change.
I would encourage your dd to talk with her friend in a reasonable nice way, if she insists she is wearing this dress, then change yours and forfeit the deposit.
I am amazed at all these people who buy dresses for £40, dresses seem to have got cheaper, not more expensive, I have had quite a few evening dresses over the years, and they were never this cheap, and I'm not even in the designer range.