Does your friend display upset when those who have lost pets talk about them having "crossed the rainbow bridge", I wonder? If not (and many of us also use the rainbow emoji in SM posts when sharing our lost companions images, or memories of them), then she's being - as another poster said upthread - somewhat precious about the whole thing.
Yes, infertility is all-consuming to those experiencing it, but at what cost to those around them as they hope for their miracle babies? You've been left wondering if you need to redecorate your own child's nursary, if you ought to explain to her the meaning of your daughter's name... where will/does it end? With you wondering if you ought to rename your actual baby, lest her name (which is beautiful) offend said friend, or other friends/family in the future...?!
I've been where your friend is, re: miscarriage, followed by "second child infertility". My youngest is both a "rainbow baby" and a "surviving twin". I have friends who have lost babies, and friends who have no clue what infertility/miscarriage is like. Sadly, it's the way it's always been - but this term of "rainbow baby" (because rainbows always follow a storm... apparently) and the pretentious self absorbation amongst those waiting for them? This is new.
And new isn't always a good thing.
To be honest, @Notmushroomleft , if I were you, I think I'd be re-evaluating this friendship and wondering if it's worth the pain and stress it seems to be causing both of you, long-term. A rainbow symbolises hope - and what is more hopeful, for you, than your daughter? Your friend, sadly, is subsumed by her grief at the loss of her pregnancy and the desperation to replace the hope which she would have felt during it. Her situation is very sad, and I'm sure that most of us on this thread can either identify or sympathise (perhaps, like me, even both) with how she feels... but that doesn't give her the right to make you question and doubt yourself with regards to your home, your daughter, your life.