Updated by NWF Renim
Posted under General
Updated by NWF Renim
Why exactly is that her official name? Cause they say so? They have zero references to source that in fact being her name.
When she first appears in the series she doesn't give a name, so they call her "Black (Kuro) Illya" to refer to her. When she does give a name she introduces herself also as Illyasviel von Einzbern, and to avoid confusion they just start calling her "Kuro" to not mix up the two. When she later transfers to Illya's school, she arbitrarily uses the name クロエ・フォン・アインツベルン (Kuroe von Einzbern) because it has "Kuro" in the name and says she is related to Illya to explain the similarity in appearance. So I'm not seeing how Chloe is actually officially her name. At minimum the use of Chloe as the name lacks an English spelling in the Japanese wiki ,where Illya and other characters (including Miyu) have their names written out in English, which would suggest there is no official English spelling to that name yet. Additionally "Kuro" is essentially the only name they use for her throughout the entire series (I haven't reread the whole thing, but Kuroe/Chloe only pops up one single time from what I'm seeing), in fact she even tells her classmates to call her "Kuro" immediately after she introduced herself as Kuroe von Einzbern to them and so all her classmates just call her "Kuro."
If you did't read the details in the "Official name" wiki link in the OP:
"Kuro" is in fact Illya's first personality that had been sealed away to allow Illya to grow up "normal" in the Kaleid Liner timeline who has taken physical form, so she is in fact also Illyasviel von Einzbern.
As for クロエ being "Chloe," in the series Blood Alone the main character is named 黒瀬 クロエ (Kurose Kuroe), and the official translation of the manga also uses Kuroe (though they do also use Chloe for クロエ when they refer to his mentor).
Updated
Going to bump this, because now there is something that can be used as material to support the proposed alias without citing a wiki with no references.
The official Japanese website for Fate/kaleid liner prisma Illya anime has the nickname for Kuro written in English as "Chlo" (ie they're romanizing クロ/Kuro as Chlo) which would thus imply her full name will be "Chloe" in English. Noticed this after recently checking their website, though at the time I had written the wiki for Kuro (fate/kaleid liner), the site originally had it written out as "Kuro."
The character page for her: http://anime.prisma-illya.jp/character/#kuro
Old: https://dl.dropboxusercontent.com/u/35308228/detail_kuro.png
New: https://dl.dropboxusercontent.com/u/35308228/detail_kuro%20%28new%29.png