I see myself in her overcoming her difficult past and moving forward.
I identify with her in how she overcomes her difficult past and moves on with her life.
つらい過去 = difficult past
乗り越える = overcome
次に進む = move forward / move on with one’s life
自分を重ねる = I see myself in / I identify with
“identify” は「突き止める」と「見分ける」の意味もあります。例えば “They identified the criminal using security camera footage.” (防犯カメラの映像で犯人を突き止めた。) 似ている英語の “identity” (アイデンティティ)もあります。