The quest is not recognizing my unicorn who fits the criteria. She was able to do these quests before she got a custom skin. The option to chose any pet is not there, it simply says "You don't have a pet which meets quest criteria!"

The quest:
Conclave of Warlocks is looking for a clever (89) Black Unicorn which is Descendant of Heroes in 3 generations for a mystic ceremony

My unicorn:

[GP3 Isabella
Intellect: 138
Genetic Power: 3
Color/Pattern: Black, Normal geneticaly.
Busy: No. She is active.

All her genes are researched. I would hate to remove then re-apply her custom skin just to do a quest. The reward is a good amount of silver at 11,500.

Am I missing something or is it the custom skin causing the trouble?

Color related quests request a specific color, single one or one from a list. A pet with custom skin has its own color, for example your unicorn now has [M]: Fast Trot Black instead of regular site color. So yes, pets with custom skins won't be accepted because questgiver wants to see a specific pet's appearance.

It is a black custom skin :/. Well fine then, picky questgiver. I will have to get unicorn just for the quest. Thanks for the answer, Angel.