Natural colorants (for example beet juice or carrot juice extract) are rare, but they do exist. These colorants, while available are very expensive and tend to be unstable, so it is unlikely you'd see them in dog shampoo.
Artificial colors aren’t “pure” single ingredients. They are multiple ingredients that are often synthesized from petroleum. Unfortunately, they are also often contaminated with byproducts from the manufacturing process.
Artificial colors have linkages to organ damage, cancer, birth defects, and allergic reactions.
Unfortunately, since there are no regulations that require a manufacturer to list whether or not they have added artificial color to pet grooming products, it can be difficult to know if it has been added to your dog's shampoo.
You can usually identify artificial colors from visually inspecting the product.
No dog shampoo is naturally hot pink, blue, red, purple, or has sparkles.
If they are listed on the label, you may see something like the following: