Does a commander lose its color identity if it changes colors on the battlefield?

Color identity is a deck-construction rule calculated before the game starts, not a property that updates during play. According to CR 903.4, a card's color identity is determined by the mana symbols and color indicators that appear on the card as printed (in the Command Zone). It is locked in at the start of the game and used solely to restrict which cards can be in your deck.

Even if your commander becomes a different color on the battlefield — for example, because an opponent cast Painter's Servant naming blue, making all cards blue — that change is a game-state effect under CR 105.3 and CR 613. It affects the commander's color as an object on the battlefield, but it does not retroactively alter color identity, which was already set before the game began.

Color identity is composed of: (1) all mana symbols in the card's mana cost, (2) all mana symbols in the card's rules text, and (3) any color indicator on the card (CR 903.4). None of these are things that can be changed by an in-game effect like a color-changing enchantment or activated ability.

Concrete example: Your commander is Rhonas the Indomitable (green). An opponent controls Painter's Servant set to red, making Rhonas also red on the battlefield. Rhonas's color identity remains green only — the color change does not mean you could have included red cards in your deck, and it has no bearing on any Commander format rule beyond the battlefield game state.

In short: color identity is a static, pre-game characteristic defined by the printed card. In-game color changes are real for purposes of spells and abilities that care about color, but they never alter color identity (CR 903.4).