The manga reveals at the end that Mitsuki's tumor did not grow while she was Fullmoon, and that's why she's still alive and managed to make it to the age of 16, when her original diagnosis was to have one year left without treatment. While she agrees to get operated, the manga never mentions anything about a medical breakthrough where her tumor could be removed without damaging or removing the vocal chords... does this mean that Mitsuki will die soon, anyway?
No, Mitsuki's name was removed from the Death List so dying young is no longer her fate.
If Mitsuki's entire motivation for becoming an Idol Singer is so that Eichi will recognise her, what's the point in changing her age, hair colour and name to do so? (This is eventually justified in the manga, but not the anime.)
In-Universe she explains that he would still recognize her because of her voice and because of the Power of Love. Though since Eichi is dead, Mitsuki is revealed to be a Death Seeker, it's likely she was lying about this and merely changed her appearance so that if she was picked in the audition, her grandmother wouldn't instantly recognize it was she.
