Encouraging people with the idea that "everyone is unique," while partially true, is but a futile attempt to claim a special niche within a world filled with billions of other humans. Though physically we are all different through DNA, and in personality through experiences of the past, we all fall into the category of human. And therefore not all of us, but rather none of us, are special at all. None of us can claim a special place amongst anyone else because we are all subject to the same human cycle: birth, life, and death.
There are names that have gone down in history moreso than others, of course. People of historical, creative, and religious importance. Presidents, writers, popes, etc. These are the names that large groups of people remember. And why? Because they did something significant; something that's lasted. Not because of who they were, but usually because of something they did. It is not their uniqueness of character that causes them to be remembered (though this is sometimes noted after they have achieved something great). It is their "uniqueness" of action.
Edgar Allan Poe would not be a revered writer (notice "writer," denoting action) if he had not written his poems and short stories. And had he not written these, his madness would never have merited him a spot among other great writers, but instead merited him a few brief glances from members of then-present day society and nothing more. Julius Caesar would not be remembered as a great emperor (i.e. ruler of an empire, denoting action) had he not taken Roman rule from the senate. He would have simply been a Roman citizen with delusions of grandeur, forgotten like all the rest of the common Roman peoples. Kurt Cobain, had he not been a musician (once again denoting action), would have been nothing more than a drug addict who mixed with the rest of the common public, doomed to die and never to be spoken of again.
If character was the determining factor for remembering and forgetting, I know and knew plenty of people whose character far outweighs that of many "great" historical people. Uniqueness, then, is not so magical a thing as some would have you believe, because unless you do something to merit you a place in history your uniqueness means nothing. As it stands, that unique quality you prize is very likely held by hundreds of other people, and many of those are likely better at or with that quality than you are. You are not special, nor are you unique. You are human. And to be human is to be the same.
You are born, and for an infinitesimally brief moment you are the youngest person on earth. Then that moment passes, and you begin the dying process. Your body will disguise this with growth, strengthening, and with the formation of memories and experiences. But all the while, you (along with everybody else) are dying.
While you are still alive, however, you will live a life parallel or similar to somebody (often many somebodies) else. You will achieve things that have been achieved before, and should you manage to break through the common crowd of birthed, dead, then forgotten humans and achieve something truly great, you will be remembered not by who you are, but by a name attached with your actions. Your character will be forgotten. In this, nobody is special. We are all the same.
Then, you will die. You will pass from this earth. You will lose the ability to remember, and the name-action pairs you did remember will no longer matter. You will cease to exist, falling into Oblivion. And though your name and actions may (depending on their greatness) may be remembered by some (possibly even many), you cannot escape the truth that has existed since the dawn of humankind as a whole:
You will be forgotten.
It doesn't matter, then, how "unique" (or how attractive, intelligent, hard-working, etc.) you are. Because unless your actions are important or great enough to last, nothing of you will survive. It is actions, therefore, that defines humanity; defines "uniqueness"; defines history; defines everything. For in the end, there is nothing else.