Human Identity

If I understand this correctly, Marx believes that in an ideal society, there is no class nor private property, ergo we are all equal, and own nothing.  If this understanding is accurate, the only conclusion I could arrive at is that in Marx's ideal society, for one to participate in it, one must throw away his/her identity as an unique individual.  What I know from my own personal experience in the society I've been living in for two decades, is that it is common for one to strive to make oneself known in society.  What I mean is that most want to be themselves and act in such a way that makes them who they are.  With Marx's society, it seems that he thinks it best that everyone basically become the same sort of citizen, acting and responding in the same way to every situation.  Wouldn't this most likely be the same as taking away one's free will, then again wouldn't the existence of free will hinder the success of this form of society?  If so then this fact would explain why this society is completely inviable, although still slightly practical.  As Marx assumes, there is no human desires present in his society, ergo no one desires free will.  With such factors in place, perhaps then one could "control" and "maintain" society so that everyone is one in the same, and so that everything belongs to everyone, yet at the same time belongs to everyone else.  However, the fact that something were to belong to someone, yet not belong to someone, would seem at a first glance as completely counter productive.  And again wouldn't Marx be asking society to submit its free will.  Such an assumption would seem accurate if one is looking at society as a whole given that said "something" is a limited resource.  However, it is possible for such a society to exist in a smaller community, such as a household, at least to an extent.  It would be true that everyone in the household would see everything belonging to one another, however, that's as successful as Marx would get in implementing his ideals.  To have a no class at all would bring about ruin to the household since no one knows what should be seen as right or wrong.  In fact, it would seem to me that one of the flaws in Marx's philosophy would be that in order for everyone to act in such a way that they only benefit the society, they must first learn what is right.  So there must be an individual of "higher class" to set certain rights, so society knows that acting in accordance to these "rights" is the only way to benefit society.  Claiming such would prove the inviablility of Marx's class-less system, and because there is an individual of higher order, then he/she must have ownership of something that puts him in such a position.  This means that within this society were there is supposed to be no private property, there is at least one individual with private property.  So because of this single flaw, Marx's ideal society collapses upon itself, because there is an individual who has claimed his identity, and become an unique individual who controls a society of individuals who lack an identity.