You can change the color of your chocobo in Final Fantasy XV by talking to the girl at the Chocobo Post in Duscae.
This is only possible after reaching story chapter 3 and unlocking chocobos. See How to Unlock Chocobos in FFXV.
The girl sells Chocobo food and lets you customize the color of your feathered friend. She even has an option for renaming your Chocobo and equipping it with medals (which will hang around the bird’s neck, for that extra touch of customization). You can win medals from chocobo races.
You obtain new dyes from side quests and chocobo races. Some are also unlocked automatically. In the customization screen press R1/RB to select whose chocobo color you want to change. It can be adjusted for each team member. They can all have a different color or the same color, whatever you prefer.