When my wife and I first got together, my stepdaughter was a bit older than your daughter, 12. We are late 30's early 40's, both out of terrible marriages, both knowing exactly what we were looking for in a relationship. We violated so many mumsnet cardinal rules, we are no doubt destined to the 37th level of hell.
Anyway, we actually moved in together at about the six month point. I didn't go out of my way to engage with her daughter as the super fun step dad guy wannabe. Didn't set out to do fun things focused on the daughter or to be the cool guy.
What I did do however was simply engage with her and include her in my day-to-day mundane life. Take mealtimes for example. I quickly became the person who did pretty much all the cooking. I know one of the common things people do is plan out a meal prep and such that is designed to be a kid friendly event and production, but yeah, I didn't do that.
Instead, I'd be working on dinner, and she'd be doing homework or what ever, and we'd just talk. Silly things about our days, stuff like that. If I needed an extra hand for something, I'd ask her. I didn't treat her like a kid, or go out of my way to put her on a pedestal or anything. Just treated her like a person.
What I found interesting though was that over time, she'd start engaging me in her mundane day to day things as well. One memory I think I will always have was she was preparing for a school art show and she sought out my opinion on organizing her portfolio. I remember sitting there with her on the floor arranging things. Nothing special, pretty mundane, just being part of each others life.
It didn't take long to discover that she and I were similar in a lot of ways, most notably being fascinated by simple things that many other people would not even notice. Things like driving down the road and noticing a hedgerow with a seemingly pointless gap in it and wondering why that gap was there. Or where we lived there was a lot of open farmland and this one spot in particular stood a lone tree that she noticed and pointed out.
I was quite honoured to walk her down the aisle when she got married, and it was a blast being the goofy dad on tik tok during the father daughter dance.
I guess my point of all of that is that though she never said it, I suspect my wife may have noticed and felt similar to how you are feeling. The thing is, we just have a different type of relationship than she has with her mother. They love each other and are very close, just in a different way.