The manga failed to deeply explore Nina’s identity crisis. It’s not so much that she did not dare to tell others who she was. The presence or absence of proof did not matter. What mattered was whether she decide to live as Nina or as the girl who was already dead. Telling others that she possesses the memories of a dead person essentially entails an identity change for her, at least in the eyes of others. She wouldn’t be able to live her life simply as Nina anymore.
In the end, she decided that she was Nina, that her lover must love her for who she now is, and her past parents and brother must get over her death, for she now had a new family she treasured too. Telling others that she was someone else in the past had no use but serve to complicate the situation.
In other words, does memory determine who a person is? If you possess Someone else’s memory, are you that person? If you lose all of your memories, are you still you? Nina essentially concluded that the answer to the first question was no. She is just a girl named Nina who somehow possesses the memories of a dead person.
While I do agree with what you said, I personally see it as memories do determine who a person is. If you wake up right where you left off but in a different body, you're still you. Regardless of how circumstances may change. I would then continue to argue that yes, if you lose your memories you are a different person starting from that day forward. I certainly have to say though this all could have been avoided if like you said, the author explored her identity crisis. Then you wouldn't be left wondering.
I tried to get into this, I really did. It's a great story but I simply cant forgive her for not just telling everyone who she is. The memories are there. She has the proof. I just cant man.