The kits are designed for use by children so they contain water-based polish. These types of polish contain dyes somewhat similar to food coloring to replicate the cosmetic-grade polish results.
To remove any leftover color, nails can be gently scrubbed with hand-washing dish soap and a nail brush. You may also try adding a small amount of baking soda to your nail brush before scrubbing. After washing, you can also dry skin lightly and apply a dab of toothpaste, then gently rub the area. Rinse the toothpaste off and then was again with soap and water. Repeat if or as necessary.