After Mina's car accident, she underwent facial plastic surgery and no longer look the same as before (in heights and sizes too, very noticeable
). Basically, there were two different actresses who took on the role Mina. Before the accident the 'original' Mina face was portray by Chae Jung Ahn. After surgery, Kim Sa Rang took the role.
The two actresses look nothing alike. To me, Kim Sa Rang is much prettier than Chae Jung Ahn. She also has a rounder face shape than Chae Jung Ahn, which makes the 'identical twin' part unbelievable. That is why Mina's boyfriend couldn't recognize her after her surgery.