Did you make it clear you were looking for a stepfather for your daughter as well as a partner before he moved in? And what your expectations and desires were in that area? And what's the rest of your relationship like? What's the rest of his relationship with HER like? Does he play with her? Talk to her? If she adores him like a father, he's clearly not totally ignoring her...
I can't help but notice there's no mention of any other elements of your relationship in your OP... it does read a little bit like you moved him in so you wouldn't have to parent alone...
"I feel like it's getting to the point where I'd be in exactly the same situation if I was on my own. I have to get her up, give her breakfast before I go to work, dress her and take her to my parents house..."
Well, yes. She IS your daughter. He CAN choose to help, and many partners of people with kids (myself included) get involved in a very hands-on way.
But others don't. The parent takes responsibility for their child and they have a relationship with their partner in order to have a relationship with a partner, not in order to have a co-parent. That needs to be discussed and made clear so everyone knows what to expect.
And although I'm now (2.5 years in) much more involved with my DSS, I don't consider myself a "stepmother" just because I live with my partner and his son. The role of 'step-parent' is a HUGE ask, and it takes a long time to get there. It doesn't just automatically switch on once you get a set of keys, and there's a massive emotional learning curve for everyone involved, no matter how well-intentioned everyone is.
He doesn't sound like he's speaking to you in a respectful way about it, and that's also absolutely not on. You should be able to ask for help if you need it, but before that it sounds like you could both do with setting some clear expectations... (and maybe posting this on the Step-parenting boards for a more balanced set of views – anything 'step' related gets WILD out here)
Speaking of which, since I'm going against the grain a bit, I'm going to duck and cover now...