Does a face-down creature have a color identity in Commander?

There are two distinct concepts here: color identity (a deck-building rule in Commander) and color (an in-game characteristic). These are governed by different rules and must not be confused.

For in-game color: CR 708.2 states that a face-down permanent has no characteristics from its printed card — it has no name, no mana cost, no color, no type, and no abilities. It is effectively a 2/2 colorless creature with no other characteristics while face-down (CR 708.2b). So during the game, a face-down creature is colorless.

For color identity in deck construction (Commander rule): Color identity is determined before the game begins, using the card's full face-up text, mana cost, and color indicator (CR 903.4, 903.4a). A card's face-down status during a game is irrelevant to this calculation. So if your card has a colored mana cost or colored pips in its abilities while face-up, those count toward your Commander deck's color identity requirements.

In practice, this means you cannot include a Morph card in your Commander deck if its face-up version has a color identity outside your commander's colors — even though it will be played face-down as a colorless 2/2.

Example: Ixidron turns all non-token creatures face-down. Those creatures are now colorless 2/2s for all in-game purposes (e.g., a blue spell that says 'target blue creature' cannot target them). However, this has no effect on deck-building legality — they were evaluated for color identity before the game started.