Being unique is great. However, the story you shared, Anutik, takes the adjective "unique" to the next level. The claim of the Sweedish parents of the poor kid that his name "Brfxxccxxmnpcccclllmmnprxvclmnckssqlbb11116" is "a pregnant, expressionistic development that we see as an artistic creation" sounds absurd within today's societal boundaries. Perhaps, on another planet, this name would be functionable and acceptable.
Not long ago, I heard about a couple in New Zeland naming their child "4real." The court or the church, whoever registers babies, refused to register this name, as it did not begin with a letter. The parents, on the other hand, claimed that the child would stand out and feel very unique, when he grows up.
Regarding the name Ilya, perhaps, the sound of it is unique in America; however, it is a real name, so it is far removed from being absurd, unlike the examples above.
"Some say the Muses are nine: how careless! Look, there's Sappho too, from Lesbos, the tenth." Plato
