Sometimes parenting roles and limits are violated on purpose, and other times they are issues of an emotional nature. This second one is where your wife seems to be.
Your wife sounds like she has some very specific emotional needs that were not met in her past. She appears to need the approval of your daughter, more as a peer than as mom-daughter. Instead of being careful about dress, makeup and similar, she violates that boundary for the desire of gaining approval from your daughter. And you...you get to play the authority figure.
I have little doubt that your wife is emotionally hurt. Where and when that hurt occurred I have no idea. But, I will say that it sounds like when she was about 12-14 years old. Why? She acts much like a pre or young teen. She treats you as the father figure and allows you to take on fully what should be a shared role of parenting. You, out of sheer frustration and even apprehension do take it on fully, figuring that if you do not do so...what on earth will happen with the kids.
As long as you are there, in this roel, she will use you (your wife) to avoid her own responsibilities and to assist her insecurities.
Without a doubt she needs some help to grow. This cannot be motivated by you, as you are the means that she uses to act out her issues. Conversations with her would be and I will bet are, deeply fruistrating. I would imagine they turn into very juvenile feeling back and forth issues, with you feeling like you are dealing with another female teen.
Until she is willing to get some assistance, this issue will tend to remain. But, if you seek out a therapist first, then ask her to go with you after session or two, this improves the odds greatly that she will go. And, once in counseling, she will be able to get past her developmental sticking points.
Know that this is not you, and the feelings you have about being placed as the head of the ax squad are very real. She has placed you there, and now must come to the plate to be an adult. This however will only occur if she is willing to look at her issues, and that is why I feel you have to set the stage and enter therapy. She will then have someone to follow, which is her pattern. Steven