Situations like this are very complicated! As you stated, you are not her mother. But it sounds like you ARE her caretaker...think of yourself as a nanny. You are taking care of other people's child. You have her at home without the parents there. You take her places she needs to go. I'm guessing you help buy her clothes and food. They aren't doing it themselves. You DO have a role.
Would a nanny/babysitter discipline? Yes, they would. Would they have the child refer to them as "mom" or anything like that? No, they wouldn't. (not saying you're having her do this)
As an adult, who is the only one around, you have the right to discippline. if there are parenting decisions (how long can she stay out at night? what clothes is she allowed to wear? what boys can she date?), you would not have say in those things. But you WOULD have the right to present the question and have an answer given back to you for you to enforce as the "nanny".
I know you're the step-parent and not a nanny, but it sounds like they aren't wanting you in a parental role. To be honest, as a teen, when my mom got remarried, it was a HUGE mistake for my mom to make my step-dad BE our dad. My dad had left, and he was allowed to fully take the role as dad and do whatever he wanted. NOT a good idea. He was (and is) NOT my father. But he is an adult, and he should have the right to have some disciplining power, especially in his own home. It's a fine, difficult line to find. I know my situation was different. He wasn't in the parental role for years prior. I was an older teen when he arrived in the picture.
i know I wouldn't want a child in my home that I wasn't allowed to discipline at all. I think that is completely unrealistic for them to expect that and maybe going more along the lines of a nanny (the type of discipline they are responsible for) would help you all be able to balance what is okay or not? Not that you would refer to yourself as a nanny. Just to help give a perspective of how to discipline without parenting.
Good luck. It sounds like a very frustrating situation! If the step-daughter wants to live with her mom, maybe she should. That doesn't need to be your burden. Her mom will learn soon enough what it's like! And good for you for parenting these past several years when no one else would.