When my ex started to see the woman who became his new wife, he would take our toddler son along on their dates. Essentially, they were playing "happy families" with him - little boy, Daddy and Mummy. Except, she's not his mum, never has been his mum, and never will be his mum (she refers to herself as "your Dad's wife") - and I say that as someone who actually has grown to like her. We're not friends... but we're civil towards one another and I'm grateful to her, mostly, for the relationship that she has built with my son - and for her understanding precisely why I was so furious when I (a) found out about the affair, (b) that my son had been used as cover, and (c) when I discovered that my ex was trying to get our son to call her "mummy".
When my son told me, upon their return every Saturday from swimming lessons/"Dad And Son Time" about "Daddy's new friend, X", the one thing I will never forgive ex for happened. He looked down at our 2 year old son and called him a liar. Daddy didn't have a new friend, X... what is he on about?!... he's lying!!! However, he saw her for another 2 and a half years - lying to me, but no longer using our son as cover, because he knew son would "out" him again - until one Saturday, he said that he'd take our son swimming. Great, I thought and duly got him ready. Ten minutes after they left, the bastard phoned me to tell me that the woman who is now married to him, had given birth to his child that morning and he was taking our son to meet "his new sister", before hanging up immediately and turning his 'phone off. Their daughter was in a SCBU at the time, and desperately ill. He took a little boy who was excited because he thought he and his Daddy were going swimming... into a babies intensive care unit instead. My son came home, but my ex and I have barely spoken since... and that was 9 years ago now.
So OP, I do completely understand your instinctive reaction - your daughter is your daughter, not this other woman's child and how dare she refer to her as such... but at the end of the day, the others who have already said this are right. Your daughter has a father. Who may, or may not have a new girlfriend. Who he may, or may not choose to introduce his daughter to. And there's nothing you can do about that, I'm afraid. What you don't want to happen is for him to put the child you share together in a situation similar to that which my ex put our son in. Because my son's earliest memory, it turns out, is his Daddy calling him a liar and me locking him (ex) out of the house... and he has a severe phobia concerning hospitals. I can't change that... but I will never forgive ex for it.
Men having affairs lie. Women having affairs lie. They don't want to be caught out, and they'll drag others into it by accusing them of being liars, shit-stirrers, jealous of your relationship with them... and so forth. Even if that "other person" dragged into it is a small, vulnerable child. They don't care, as long as they're not found out. As long as they're not discovered to be the lying lowlives that they actually are, how they make their previously loved ones feel - doesn't matter one bit.
If this relationship lasts - and there's every chance that it will, I'm afraid - then you need to lay down the line in the sand which you will not tolerate being crossed, for the sake of your daughter. Mine was my ex's new wife being called "Mummy" by my son. Not entirely because of my primal reaction to it (it hurts, being replaced in every aspect), but because of how confusing my son found it. On their wedding day, he was 5 years old, and not only was he not invited to it at all, but he asked me if I would still be his Mummy now that Daddy was marrying X. That's also something I will never forgive ex for. Our son had stewed on this for weeks, before he broke down in tears and it all came out. And then I simmered for a week until they were back from their honeymoon, and made my feelings very clear. But I did so civilly. And that's the key. Civility. Be the better person, OP. Rise above their shitty behaviour and look down your nose at them. Become excellent at weary sighs when people ask you (and they will) about the situation - you're so tired of having to explain the immature actions of your ex and his bit on the side... why don't they (the nosey people) ask them about it? It's nothing to do with you. Oh, but by the way, daughter's name is Y, not X...!
It'll hurt like hell for a while, and you'll feel like your insides are being clawed out - because you're frightened about the way in which your child is being dragged into this mess of her father's creation. But you will come through it. And you will always be your little girl's mum. No one can ever take that away from you, OP, no matter how hard/much they try to.
