Artificial lashes are made of a hypoallergenic synthetic polymer. Its scientific name is polybutylene terephthalate (PBT). Silk, Mink and other terms are simply names that were made up by manufacturers to classify materials by width and gloss — animal fur isn't used in any of them. They are outdated, and if an artist uses them, you can clearly see they are either unexperienced or haven't gone to training to upgrade their skills for quite some time.
There are several steps in making artificial lashes. First, thin long hairs with different thickness (from 0.03 to 0.3 mm) are formed from a black synthetic mass. Then those hairs are fixed on a sticky tape and cut into different lengths with a sharp blade. Then they are reeled on a roller of a certain diameter to form a desired curl and sent to specialized high-temperature chambers. Then they are taken out, left to cool down and put into the chamber again so the curl stays on for several years.
Colored lashes are made with a special hypoallergenic pigment, so artificial lashes themselves can't be a source of your client's allergy.