In Commander, does color identity prevent me from taking control of a different colored permanent?
No, stealing an opponent's creature of any color is fine.

The only limitations are the ones explicitly imposed by the rules:


Color identity applies during deck creation. You may not put a white card or a card that uses white mana symbols in your red/black Olivia deck. (You could include a black card that creates a white token, though; or an extort card, since that's just reminder text.)
(Historical only.) Before 2016, there was a special rule limiting how you generate mana: mana outside of your color identity would become colorless when it entered your mana pool. So if you were playing Olivia Voldaren and you would add {W} to your mana pool, you'd get colorless mana instead. This meant you would might trouble activating the abilities of some of the creatures you stole with Olivia.