Can a person play the piano by following a person's voice such as pitch and octave?

I'm not sure what you mean by octave, however if someone is singing a melody a pianist who can play by ear can pick up the key which the singer is in. An octave is every eighth tone, e.g. C to C up or down. It has nothing to do with playing the melody unless the melody is being played in octaves instead of single tones.

Yes, but you need to train your ears first and it helps to practice. 

Agreeing with Mordent - often, since we are both educated professionals.  Either from natural ability and practice, or through ear training education, any decent pianist can play play, without error, the melody anything a singer has sung.  And we can *find* a singer when they have, through their own error, wandered into another key.  This is easiest to do when that singer is performing some kind of pop song, where the harmonic accompaniment is what they need;  although I am a professional flutist,  I also have degrees - in Music Theory - that *required* me to study piano as my declared major instrument (there is some sense in that, although I was grudgingly indignant about the time it stole from my flute study!) so I also worked for years as a piano accompanist for choirs and solo singers.  The ongoing nightmare of singers swanning in and announcing that on *that day*, they needed me to play the entire art song down a third, etc. was dependent upon the complexity of the accompaniment - I usually could supply them with *something* to sing with, but some literature -impossible to do on the fly.  Of course, this was before the invention of SOFTWARE that will do it for you instantaneously - we had to do this all in our heads an fingers. 

ANYONE with training should be able to replicate or write down what they hear a monophonic instrument, or a singer, perform - within reason of course - but we periodically hear of people that heard ONE debut performance of a new opera, and then 10 years, later, could replicate it ALL on the piano.  And I had a fellow undergrad student who would astound us all by score-reading at the piano, huge orchestral works, and get every note from all the various transposing instruments, in a massive score.  And he was humble about it - would just say that he was lucky that it came easy to him.  The rest of us mortals despaired of ever reaching THAT level . . . but THAT is not what I think your question was about.  The bulk of us, with some education, can play by ear, and play from notation, to the limits of our technique on our instruments.  I cannot play the incredible piano solo works that my brilliant husband can - but he knows I have a FAR better ear than he does, for the utilitarian piano things. 

yes havent you heard of playing by ear , guess your not a musician are you !!!!

I'm assuming you mean playing in unison with a singer?

Someone who can play the piano could do that (providing the singer was in tune), although it would be an odd thing to do.

If that's all a person can do though then they can't “play the piano”.

Assuming the person is singing notes that a piano can play (and not "E 1/4 flat") then sure - it's actually pretty simple as a voice is essentially monophonic.

I do not think so.

There are a few people who can play "by ear."  But most of us had to learn the old fashioned way by taking lessons - learning the scales first, etc.