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

In Commander, a card's color identity is determined by the mana symbols and color indicators found anywhere on the card — but this refers to the printed card, not its current in-game state. Color identity is established before the game begins and never changes during play (CR 903.4).

A face-down morph creature is a 2/2 colorless, nameless creature with no mana cost, no abilities, and no subtypes while on the battlefield (CR 708.8). However, this in-game representation does not affect color identity, which is a deck-construction rule assessed from the printed card face.

So when asking 'can this morph card go in my Commander deck?', you look at the card's actual colors and mana symbols (front face), not its face-down state. A morph card with a red mana symbol in its casting cost has a red color identity and requires a commander that includes red — regardless of the fact that it becomes colorless when turned face-down.

Concrete example: Blistering Firecat has a red color identity (its morph cost and card frame involve red). While it is face-down on the battlefield it is a colorless 2/2, but it still requires a red-inclusive Commander deck. You cannot sneak it into a mono-blue deck just because it 'looks' colorless face-down.

In summary: the face-down state is irrelevant to color identity. Color identity is a static property of the printed card used only for deck construction, governed by CR 903.4 and CR 708.8.