Same name
I haven't seen the film all the way through, and I don't know how old they were when their mother gave them up, but surely their births would have been registered at the same time, and so would hardly have been given the same name.
shareI don't know how it works everywhere, but I in some places, I believe the child doesn't have an actual official name until the parents give them one on their birth certificate. All they are probably going to be known as, until that point, is something like, "Baby Boy or Baby Girl and what ever the last name is," or just by their gender and the name of the mother. The latter one is how a child, who I know about, was identified in the hospital. The child's gender and the mother's name were on the child's id bracelet. In a situation of adoption, the adoptive parents will name the child. If the adoptive parents didn't know that the other had a twin, they probably wouldn't know about anything like the other birth being registered.
Also, I know of a situation, where parents knowingly gave two of their children similar names. They were a family that my mom knew while growing up. This family had a child name John and one named Johnny (not sure if this is how they spelled Johnny).
Then you can have a situation where a parent inadvertently gives a child the same name in different forms. For example, I know someone who gave their children names that they later found out were forms of the same name in different languages. I won't say what they named their kids due to privacy reasons.