I have to say the scenario you describe reminds me of a former sister-in-law and how she felt about the birth of my own daughter. She had waited 10 years after getting married to have a child because her husband did not want children, only agreed to one child in order to please her and told her that the baby would be HER baby and he would take no active role with the child. He meant every word of it, sorry to say.
The sister-in-law was herself the youngest of 8 children and grew up believing she had to constantly compete with her siblings for attention. Add in the fact that Grandma was always blatant about showing favoritism, and the youngest grandchild was always the apple of Grandma's eye. Now you understand how important it was to my former sister-in-law to have the youngest grandchild.
But one thing she didn't think about was the age difference between her and me. I was much younger than my husband, hence 12 years younger than her.
My sister-in-law's first two pregnancies ended in miscarriages. A baby, understandably, became even MORE important to her. Sensitive to her keen desire to have a child, and considering that she and I were good friends at the time, I did let her know when my husband and I decided to try for a child of our own. She seemed to handle that news well enough at the time.
Shortly after I told her that, she began her third, and successful, pregnancy. One month before her baby was born, I got pregnant. When her baby was a few weeks old, my pregnancy test was positive. My daughter is 8 months younger than her daughter.
Man, did her attitude toward me ever change the moment I told her I was pregnant. I attributed her horrible reaction to the fact that she was a new mother herself and had many overwhelming life changes to deal with, but I was wrong about that being the reason why she suddenly treated me like dirt. I was very shocked by the horrible things she said about me and my baby. The other family members were very shocked as well. She sent hate letters. She slashed a tire on my car. She came over with a pellet gun and shot out a window. When my daughter was a few weeks old and was sick for the first time (every mother knows how terrifying the first illness is to an inexperienced mother - I was very frightened) my sister-in-law called me to say Grandma had told her my baby was sick and that sister-in-law hoped my baby would die because after all, I only had the child in order to have one younger than hers. Can you believe it?
End of friendship between that sister-in-law and me. Period.
Now, her attitude was extreme and frankly, I look at the whole of it and the whole of the other issues in her life, both then and now, and I suspect she's got some serious mental health issues - but I'm not qualified to judge such a thing.
Here's what I want to say to you, though. Perhaps your sister-in-law didn't choose her birth date for the reasons you believe she did. Can you find a neutral manner to ask how she chose that particular date? Perhaps the fact that her twins get all the attention and your daughter gets none on the shared birthday is because she lives where the rest of the family lives and the family gets to interact with the twins all the time anyway? Does your daughter get even a card or a call from her father's side of the family? Is the family's reaction to your daughter's birthday the same as it is to your other children's birthdays? Is their reaction to her birthday the same as it was before the twins were born? Do the relatives make a great big fuss over going to the twins' birthday party?