I truly don't think Aya really became like her father.
She may be a little nuts but who wouldn't after everything that's happened.
Aya is too kind and compassionate to become a psychopathic murderer. When Maria said she became like her father, I think she meant she became a doctor and her melancholic look was merely her just reminiscing about the man she loved.
If Aya became like her father, she would end up betraying her mother and the people who sacrificed themselves to help her. She seems to have a need to please her mother, so I highly doubt she would become a murderer.