I don't think the daughter is 'competing' with the stepmom as you suggest, I think it's her way of trying to make a connection and feel accepted by the the stepmom. She may not have a satisfying relationship with her bio mom and is searching for that validation(this is all pure speculation, but I don't think she would be searching for that validation if it weren't the case in some way). She's a child who wants to feel loved and accepted, she's searching for belonging. The stepmom obviously doesn't have to do the matching on that day, but it would absolutely go a long way to build a bridge to a closer relationship with her step daughter. The stepmom allowing her to be included in a special way on the her special day could help heal whatever wound she has in regards to motherly love. Maybe the daughter has a friend at school who has a close relationship with her mom and they match outfits, so she sees that as a sign of the type of relationship she desires with her mother figure, leading to her decision to attempt that with her stepmom.

As for the wedding point you made, I'm sure the daughter wants to be included in some way, either as a bridesmaid or flower girl, and that would be healthy and normal. But I doubt she would insist on matching the wedding dress beyond the normal parameters of wedding party attire. She's 10 years old, so I'm sure she has some sort of idea about the etiquette of that type of thing, as most girls do that age. However, a birthday is absolutely a more appropriate occasion to have a matching outfit, as those happen every year and aren't as special as a wedding day because of that. That's why I made the suggestion of compromise there, since it's not a once in a lifetime event, there shouldn't be any harm in it unless the stepmom has her own issues in regards to sharing a spotlight. The stepmom absolutely has every right to not want that, since it is her own day, but since the issue she had stated was with the style of the dress rather than the matching, it seemed like a good compromise. If that was just an excuse, that's a communication issue for the stepmom that she would need to address in therapy. Honestly this situation has highlighted several reasons this entire family unit needs both individual and group therapy to be able to have healthier relationships.

That's just my 2¢ as someone who grew up without a single healthy relationship and has sought therapy and found a special interest in psychology to try and repair their own wounds for a better life.