Multicolored horses... Great! But why no multicolor chocobos?

[Enkidoh]

I presume you haven't played another FF game based on your above post, in which case, chocobos are basically Square Enix's mascot (and the mascot of the FF series in general) - in every FF game horses are virtually non-existent (often the only times a horse shows up is either as a generic enemy, like the so-called 'Crazy Horse' monster in FF1, or Odin's steed Sleipnir), and accordingly FFXIV keeps to that tradition.

Most FF games allow the player to ride a chocobo as a mount, whether just as a rental, one they have to catch in the field, or can actually breed their own at a stable, it all depends on the FF title. This page has all you need to know about chocobos: http://finalfantasy.wikia.com/wiki/Chocobo

Lore wise for FFXIV, horses are actually native to the Far East (the continent of Othard), and are not found naturally in Eorzea, other than a species of unicorn being native to Coerthas, hence the player mounts. The special event Heavensturn that happened at the start of the year fittingly dealt with why horses were not well suited to Eorzean conditions and why chocobos are preferable, albeit in a humorous way.