OP the advice above about thinking of yourself for the moment as a step parent is good, and you may find it useful to read some of the step parenting board and talking to them.
It is a very hard situation for you: you think of yourself as your dd's parent, you want to be an equal parent and for her to treat you as her parent, you want her love and trust the way you see her give it to her mother, you want to be part of shared parenting in a parenting approach you and your wife both agree on, and you've loved and thought of dd as your child since she was born.
Your dd's and wife's reality however isn't the same. Your dd has experienced single parenting and your wife has been doing single parenting. In your dd's world, suddenly a new adult has moved in, is changing things, wanting to be part of things, challenging mum about how things are done, taking up time and territory that she thought of as hers and her mothers - she may well be quite angry with you about the way her little world has changed, and she may not yet see the changes are all good ones. Your dd isn't an adult, she doesn't have that internal relationship with you, she has no certainty that having appeared suddenly in her life you won't disappear again, and you're going to have to get to know her, build trust and earn that place in her life in very small steps.
I can imagine how upsetting that is to think of, but that is the reality. You would probably find it works better in the long term to approach this like a step dad moving in to an existing family for the first time, and finding out how other men have done this.