I can relate to what you're saying, OP, because it sounds very similar to the situation I found myself in 20 years ago when my DD was born.
I was 20, had moved back in with my parents to have my DD (her biological father was very abusive and I walked in fear of my life, just before I discovered that I was pregnant with DD). My mother took over, completely. Like your DM, mine would refer to my DD as "ours" even whilst I was still pregnant. She insisted on being at the birth and got very shirty when I refused to allow her to name my DD with some weird made-up name (Jances) that mashed my mother's first and middle names together into one. My mother pretty much isolated me from my friends completely both before and after my DD was born - she wouldn't watch her in an evening so that I could go out for an hour or so with my friends, because my DD was born with a few issues (easily treatable, and a ludicrous excuse, really, in hindsight, because my mother had been a paeds nurse for years!). She also insisted on involving my abusive ex and his equally abusive/controlling parents in my DD's life against my wishes - and I felt completely unable to stand up to her. Because I'd spent my entire life being too frightened of my mother to do so.
Even today, years after DD and I moved out, she still feels more like my younger sister than my daughter. My mother still wields her influence over my DD (because they're actually more alike than I particularly care for!) - but not my DS (11) who was born 6 years after I took my daughter and walked away from my parents home. When I told her that I was pregnant with my DS, she sneered "I didn't even know you were having sex!" - when she knew that I'd been in a relationship with his DF since my DD was about a year old (he was the only friend who stuck by me when she was busy isolating me whilst I was pregnant with my DD, so that she could control me). She did try to take over with my DS, actually, by attempting to bully his DF into doing what she wanted rather than what we wanted for our baby... but my DS' dad stood up to her. Consequently, she loathes him and now goes out of her way to make snide comments about him and his parents in front of my DS at every given opportunity (needless to say, she has very little relationship with my DS... and, unfortunately, his DF and I split because I'd had enough of everyone trying to make me tow their line rather than the one I needed my DC and myself to follow).
My advice, OP is to pick up your DD and run. Find your own line - whether it's with your DP, or not - and stick to that. Because, believe me, if your DM is even remotely like my own, this situation will only get worse! If you want your DD to be your daughter and not your DM's, then you need to find your spine, find your line and stick to it! Being a mum is about doing what is best for our children. Setting a good example to them. And standing up for not only them, but ourselves, too!
And, actually, as much as I adore my DD - I do wish that I'd found my own spine and stood up for my rights as her mother whilst I was pregnant with her, not when she was 2 years old. I wish that I'd listened to the people who said to me what I've just "said" to you. To the family GP who had spent years - from the time I turned 16 - urging me to get as far away as I possibly could from my mother before she destroyed me completely. To my grandparents. To my oldest brother who had run as far away as he possibly could, as soon as he could. Even to my own father (who actually enables her). But I didn't. And I will regret that for the rest of my life. Because I lost out on so much of my DD's early childhood - and later...
Please don't let that happen to you and your DD.