Are you sure she wasn't just remembering seeing The Parent Trap when she was a kid?
It was pretty similar! Though it didn't have the same ending.
I can't remember their names but EM went to the same summer camp for all of her childhood.
She gets there at age 12 and greets her friend Anne (let's say), who she's known for years. The girl says, "I'm not Anne but you're the third person to call me that. Who's Anne?"
EM takes new girl, Jane (all names made) to meet Anne. They are agog. They look exactly alike. They are both adopted. They become friends and compare notes all summer.
They both live near NYC and after camp is over they meet each other in Manhattan. One of them has snooped through her parents papers and they have the name of the woman who gave them up for adoption!
She works in the Empire State building. They know what she looks like. They decide to confront her at work.
At the end of the workday, the woman comes down the elevator to the lobby and she sees these two twelve year old girls who are identical. Who look exactly like she did when she was that age. She knows exactly who they are. Her face turns white, and she turns on her heel and goes back to the elevator. She refuses to ever meet with them.
But the two sisters found each other and remained close.