Yes. Yes there is. Tecknically, it's a pink or blue unicorn. Not purple

The pink and blue unicorns have no given names. Therefore, you can call them whatever you want. Most people will just call them by their color. Like you did in your question, they would be called the blue Unicorn, and the pink Unicorn.

charlie has no friends so far in the show but if your referring to the pink and blue unicorn well, they don't have names so far they haven't even said wether they are boys or girls

The unicorn's name in Blizzard on the Blue Moon is Moonbeam.

chocolate is amazing!