Потому что у вас не edge ngram, a обычный ngram. У вас вот что индексирутся:
POST test/_analyze
{
  "text": ["Рябов Артур Артур"],
  "field": "localizedName.name"
}
{
  "tokens" : [
    {
      "token" : "р",
      "start_offset" : 0,
      "end_offset" : 1,
      "type" : "word",
      "position" : 0
    },
    {
      "token" : "ря",
      "start_offset" : 0,
      "end_offset" : 2,
      "type" : "word",
      "position" : 1
    },
    {
      "token" : "ряб",
      "start_offset" : 0,
      "end_offset" : 3,
      "type" : "word",
      "position" : 2
    },
    {
      "token" : "рябо",
      "start_offset" : 0,
      "end_offset" : 4,
      "type" : "word",
      "position" : 3
    },
    {
      "token" : "рябов",
      "start_offset" : 0,
      "end_offset" : 5,
      "type" : "word",
      "position" : 4
    },
    {
      "token" : "я",
      "start_offset" : 1,
      "end_offset" : 2,
      "type" : "word",
      "position" : 5
    },
    {
      "token" : "яб",
      "start_offset" : 1,
      "end_offset" : 3,
      "type" : "word",
      "position" : 6
    },
    {
      "token" : "ябо",
      "start_offset" : 1,
      "end_offset" : 4,
      "type" : "word",
      "position" : 7
    },
    {
      "token" : "ябов",
      "start_offset" : 1,
      "end_offset" : 5,
      "type" : "word",
      "position" : 8
    },
    {
      "token" : "б",
      "start_offset" : 2,
      "end_offset" : 3,
      "type" : "word",
      "position" : 9
    },
    {
      "token" : "бо",
      "start_offset" : 2,
      "end_offset" : 4,
      "type" : "word",
      "position" : 10
    },
    {
      "token" : "бов",
      "start_offset" : 2,
      "end_offset" : 5,
      "type" : "word",
      "position" : 11
    },
    {
      "token" : "о",
      "start_offset" : 3,
      "end_offset" : 4,
      "type" : "word",
      "position" : 12
    },
    {
      "token" : "ов",
      "start_offset" : 3,
      "end_offset" : 5,
      "type" : "word",
      "position" : 13
    },
    {
      "token" : "в",
      "start_offset" : 4,
      "end_offset" : 5,
      "type" : "word",
      "position" : 14
    },
    {
      "token" : "а",
      "start_offset" : 6,
      "end_offset" : 7,
      "type" : "word",
      "position" : 15
    },
    {
      "token" : "ар",
      "start_offset" : 6,
      "end_offset" : 8,
      "type" : "word",
      "position" : 16
    },
    {
      "token" : "арт",
      "start_offset" : 6,
      "end_offset" : 9,
      "type" : "word",
      "position" : 17
    },
    {
      "token" : "арту",
      "start_offset" : 6,
      "end_offset" : 10,
      "type" : "word",
      "position" : 18
    },
    {
      "token" : "артур",
      "start_offset" : 6,
      "end_offset" : 11,
      "type" : "word",
      "position" : 19
    },
    {
      "token" : "р",
      "start_offset" : 7,
      "end_offset" : 8,
      "type" : "word",
      "position" : 20
    },
    {
      "token" : "рт",
      "start_offset" : 7,
      "end_offset" : 9,
      "type" : "word",
      "position" : 21
    },
    {
      "token" : "рту",
      "start_offset" : 7,
      "end_offset" : 10,
      "type" : "word",
      "position" : 22
    },
    {
      "token" : "ртур",
      "start_offset" : 7,
      "end_offset" : 11,
      "type" : "word",
      "position" : 23
    },
    {
      "token" : "т",
      "start_offset" : 8,
      "end_offset" : 9,
      "type" : "word",
      "position" : 24
    },
    {
      "token" : "ту",
      "start_offset" : 8,
      "end_offset" : 10,
      "type" : "word",
      "position" : 25
    },
    {
      "token" : "тур",
      "start_offset" : 8,
      "end_offset" : 11,
      "type" : "word",
      "position" : 26
    },
    {
      "token" : "у",
      "start_offset" : 9,
      "end_offset" : 10,
      "type" : "word",
      "position" : 27
    },
    {
      "token" : "ур",
      "start_offset" : 9,
      "end_offset" : 11,
      "type" : "word",
      "position" : 28
    },
    {
      "token" : "р",
      "start_offset" : 10,
      "end_offset" : 11,
      "type" : "word",
      "position" : 29
    },
    {
      "token" : "а",
      "start_offset" : 12,
      "end_offset" : 13,
      "type" : "word",
      "position" : 30
    },
    {
      "token" : "ар",
      "start_offset" : 12,
      "end_offset" : 14,
      "type" : "word",
      "position" : 31
    },
    {
      "token" : "арт",
      "start_offset" : 12,
      "end_offset" : 15,
      "type" : "word",
      "position" : 32
    },
    {
      "token" : "арту",
      "start_offset" : 12,
      "end_offset" : 16,
      "type" : "word",
      "position" : 33
    },
    {
      "token" : "артур",
      "start_offset" : 12,
      "end_offset" : 17,
      "type" : "word",
      "position" : 34
    },
    {
      "token" : "р",
      "start_offset" : 13,
      "end_offset" : 14,
      "type" : "word",
      "position" : 35
    },
    {
      "token" : "рт",
      "start_offset" : 13,
      "end_offset" : 15,
      "type" : "word",
      "position" : 36
    },
    {
      "token" : "рту",
      "start_offset" : 13,
      "end_offset" : 16,
      "type" : "word",
      "position" : 37
    },
    {
      "token" : "ртур",
      "start_offset" : 13,
      "end_offset" : 17,
      "type" : "word",
      "position" : 38
    },
    {
      "token" : "т",
      "start_offset" : 14,
      "end_offset" : 15,
      "type" : "word",
      "position" : 39
    },
    {
      "token" : "ту",
      "start_offset" : 14,
      "end_offset" : 16,
      "type" : "word",
      "position" : 40
    },
    {
      "token" : "тур",
      "start_offset" : 14,
      "end_offset" : 17,
      "type" : "word",
      "position" : 41
    },
    {
      "token" : "у",
      "start_offset" : 15,
      "end_offset" : 16,
      "type" : "word",
      "position" : 42
    },
    {
      "token" : "ур",
      "start_offset" : 15,
      "end_offset" : 17,
      "type" : "word",
      "position" : 43
    },
    {
      "token" : "р",
      "start_offset" : 16,
      "end_offset" : 17,
      "type" : "word",
      "position" : 44
    }
  ]
}
Кроме того, вы фразу в поиске тоже через анализатор прогоняете. Я не знаю, что вы этим хотите добиться, но результат, который вы получаете, вполне соответствует тому, что вы делаете.