That would depend on the store's return/refund policy, which for Apple Store is usually 14 days.
Apple Care is like an extended warranty. It covers some hardware defects, battery issues, stuff like that. And it provides a reduced price for repairing broken screens, lost devices (if you get the Theft and Loss option), etc. But it doesn't cover color swaps.
More info on Apple Care - https://www.apple.com/support/products/iphone/
If you've already owned the phone for 6 months, you probably can't just go in and swap it for a different color. You should be able to trade it in for store credit, but you'll have to cover the rest of the cost of the new phone.