The little prince's flower is indeed unique in all the world. The most important reason why is that she is the only flower in the universe that specifically loves the prince. Due to her vanity and pride, the prince at first disbelieves that she loves him. However, just before he leaves, she apologizes for being silly, says that she loves him, and blames herself for his not knowing all along, as we see in her lines, "Of course I love you ... It is my fault you have not known it all the while" (Ch. 9). Hence, one reason why his flower is unique is that she loves him.
Another reason why she is unique in all the universe is her personality. As stated above, she is indeed very vain and prideful. Her vanity makes her say and do silly things, such as say that her thorns can fight off tigers, even though she has never met a tiger, and that she is used to colder climates, even though she has never known any world other than the prince's. In addition, her vanity makes her ask for a screen to shield her from the draft even though she is a flower and meant to be out in the draft. She even asks to be put under a glass globe at night. In addition, she would even prefer to act as if the prince was wrong rather than confess to her own wrongdoing, as we see in the line, "Embarrassed over having let herself be caught on the verge of such a naive untruth, she coughed two or three times, in order to put the little prince in the wrong" (Ch. 8). Personality traits and character traits are what make a person unique. While others may have similar traits, only the rose would think and act just as she does, making her unique in all the world, just like an individual person is unique in all the world.
At first, the Little Prince believes his flower is unique. Then, the sight of the rose garden first leads the prince to believe that his flower is not, in fact, unique. However, with the fox's guidance, the prince realizes that even so many similar flowers cannot stop his own rose from being unique.
I believe this is an analogy for relationships everywhere between people. Think of it this way: a boy meets a girl and the girl becomes special to him. He may go to a party and see many other girls that are tempting to him who look equally attractive. However, after much thought, he realizes that HIS girl is special because she loves HIM.
I love this quote in the book,
"Men have forgotten this truth," said the fox.
"But you must not forget it.
You become responsible, forever,
for what you have tamed.
You are responsible for your rose. . ."
"I am responsible for my rose,"
the little prince repeated,
so that he would be sure to remember.
I believe that you become responsible for upholding the relationship with those that you love (or, as Antoine de Saint-Exupery says--those that you have "tamed").
Thus, the Little Prince's rose IS unique because she loves him and he loves her. This makes her different from any other rose in the garden.