Why does Yukine have three different names?

Question

When Yato named Yukine, he gave him three names - Yuki, Yukine, and Setsuki.
What's the point of the different names?

Answer 1

The character 雪 ("yuki" in kun'yomi), has an alternate reading (on'yomi), "setsu."

Sekki (雪器) is a combination of this alternate reading with the character 器 (ki).

His actual true given name is "Yuki." (Yato's true name is Yaboku, 夜卜, using the kanji 卜 instead of the kana)

His weapon name is "Sekki." (the "tsu" runs into the next consonant, making "Sekki," instead of "Setsuki") The gods all seem to give their weapons names using the alternate readings and the "-ki" suffix (e.g., Sekki, Hanki, Chouki)

The common (nick)name he goes by everyday (e.g. Charlie, instead of Charles), is "Yukine," with "-ne" (音) being the suffix Yato gives to all his Regalias (e.g., Yukine, Tomone). Tenjin-sama uses "-yu" (喩) (e.g., Tsuyu, Mayu, Nayu). Kofuku uses the prefix "dai-" (大) (e.g., daikoku).

In folklore, one's true name should not be given out easily, as it reveals the person's true nature. Therefore many people take a common name similar to their true name. The weapon name is somewhat akin to a stage name, and is typically used when conducting business-related activities.

Answer 2

Each name is for a facet of the Shinki.

The Shinki as a tool.

The Shinki as a "person".

In the specific case of Yukine, the names are:

Yuki - The japanese pronunciation of the Shinki ideogram.  
Yukine - the full name of the Shinki in human form.  
Setsuki - the name of the Shinki in weapon form.

The wikia page has the breakdown of the names including the Kanji play they do with the Japanese and Chinese readings.

Kanji | Japanese | Chinese  | Human         | Weapon
      | Reading  | Reading  | Name          | Name
雪    | Yuki     | Setsu     | Yukine (雪音) | Sekki (雪器)

Also, the names are the magic words to trigger the Shinki transformation to and from person.