You're concerned because the daughter, with a temporary license, was driving while her mother was in the car? If that's the case, I would not be concerned about that. The mother is supervising. The daughter is going to be driving quite carefully. If anything, her driving may be safer than the driving of an older teen or even some adults with a full license.
I'm a bit confused by your first paragraph. It is against the law for someone with a temporary license to drive anyone without a licensed driver in the front seat with them. In Oregon, a teen with a license, still isn't allowed to drive other teens without a licensed adult driver also in the car for a period of time. I don't know how long.
I have mixed feelings about this law. For me and without the law, I would allow my teen to ride with a licensed teen if I believed both my teen and the other teen were safe and had shown that they made good decisions in other areas of their life. So much would depend on the personalities and "track record" of both.
And, yes, I would let my child get their temporary license at 15 1/2 if they were responsible in other areas of their life. I'd require good grades and having seen instances of good decision making.
This is just one more step in the maturation process. I would have rules that gradually allowed my child to become more independent. With a permit, I would first not allow anyone else in the car accept me or another adult that was teaching her to drive. The next step would be to allow other passengers. Once she had the license, I would not allow other teens alone in the car until I was convinced that it was safe. And even then, I would only allow use of the car for transportation. No just riding around.
I wonder if your daughter is mad at you because, in part, because you're so upset. I suggest that we have an easier time with teens when we're able to be calm and non-judgmental while we give them rational reasons for our concern. When we have a discussion and don't issue an angry ultimatum.
When our teen reacts emotionally, it's so easy to become emotional ourselves. Know that you have the responsibility to set boundaries and set them without falling into the drama of the teen. Be firm while being unemotional.