To be honest, I understand how you feel.
This is your FIRST baby. It is your friends second baby. She already has a firstborn. She has usurped the name you have chosen for your little girl. To you, this makes your baby less precious, and less special in your friends eyes, she does not even have a name "of her own". It is "just another Charlotte". Not THE Charlotte. I get you.
Let me tell you something.
We chose a really unique name for our second son. It is not an English name, it is a Norwegian name. But it sounds nice in English. (I am Norwegian)
One of my neighbours were pregnant at the same time as me, but we did not know eachother, she had recently moved in.
When I came to the registry office to register my babys birth, the registrar did a double take. She said it was most unusual that such a unique name were given a baby with just a few days between, it blew her statistics totally.
It turned out that my neighbour had given HER baby the same name as MY baby. Her baby was born two weeks after mine, but she registered his birth earlier. Her grandmother was Norwegian, and they had researched online to fine a nice name that would honour her late grandma, and they ended up with this name quite by chance.
As it happened, it did cause ONE problem. Us living on the same road, she in nr 92, me in nr 72. Our GP practice would forever get the babies mixed up, the secretaries would change the addresses and mix up their notes constantly. Anyhow, we became friends. The name was never an issue. We both had our reasons for chosing the name, and they were not the same. They were not the same children, but they were both blonde little boys with blonde mums and dads.
I dont think it is worth falling out with your friend, it IS possible for two friends to have the same name for their kids, without confusion.
It happened in our ante natal classes, but with years between. There is one Tim who is 9 and another Tim who is 6. It does not matter. The mum of Tim the first is feeling quite honoured that her friend chose the same name for her precious baby.