Double Date by Rosamond du Jardin (1951)

Quiet, sensitive Penny Howard has always tried to be as much like her vivacious twin Pam as Pam wished--wearing the same clothes, and letting Pam arrange dates and choose their activities. But as the girls start their senior year at a new high school, in a new town, Penny' rebellion grows and grows, and results in a private Declaration of Independence. To her surprise, she finds herself quietly cheered on by their mother, their grandmother, and finally and most happily for Penny, by Mike Bradley, the boy she was afraid Pam had chosen for herself. (from the back cover)