Depends how specific you're talking about. If you want something like, say, "a large brown mark on the center of the chest", as far as I know there's no way to ensure that. Often the placement of such markings isn't even genetic - if you cloned an orange-and-white cat, their clone would be orange-and-white, but the positioning of their orange patches would be completely different, just like how identical twins have different fingerprints.
However, if it would work to just have one strain have purplish marks while another has brown marks or something like that, that's readily doable.
Possible genetic bases for distinctive markings for humans:
Port-Wine Stains - reddish or purplish birthmarks caused by a vascular anomaly, usually located somewhere on the head or neck. Can be the result of certain genetic syndromes, but non-syndromic port-wine stains have been linked to mutations in the GNAQ gene(1). These mutations have only been reported in heterozygous form, so it's uncertain what impact they'd have if homozygous, but Basset Hounds' short-limbed dwarfism mutation is homozygous lethal and that hasn't stopped humans from turning it into the basis for a domesticated breed. If homozygous GNAQ is lethal, they'd have to have individuals with and without port-wine stains in the same families, but might be more inclined to actively show off the ones with the birthmark.
Freckles - Small brown dots on the skin that form in response to sunlight exposure, freckles are associated with polymorphisms in the MC1R gene (2). The same alleles that cause freckles also tend to cause lighter skin tone and reddish hair. Freckles are associated with slightly higher risk of melanoma, but not a serious cause for concern. These polymorphisms are commonly seen in homozygous form, and are especially common in certain ethnic groups such as Celtic people.
Waardenburg Syndrome - Waardenburg Syndrome is associated with a distinctive facial appearance, patchy depigmentation in hair, eyes and/or skin, going grey early, and deafness. 762
It's caused by MITF mutation. In heterozygotes, it can often result in no impairment in hearing, just an unusual appearance, but homozygotes have more severe symptoms(2). Still, you could conceivably have a noble house who is fine with having deaf servants breed them to be homozygous for one of the less severe mutations that causes Waardenburg Syndrome. They'd probably issue commands using sign language, and might have the bonus of being less likely to have servants passing information to their enemies. (Some slaves historically had their tongues removed for a similar purpose.)
There are many other potential genetic variations associated with distinctive pigmentary variations, far too many to list here. Some cause other symptoms, but anything that doesn't stop them being put to work or that only affects a minority of individuals wouldn't necessarily rule them out. After all, plenty of domestic animal breeds have been selected for features that cause impairments in their natural functioning, such as short nose syndrome in dogs or even the polled (no horns) trait in cattle (while cattle owners often prefer cattle without horns and will remove horns on breeds that grow them, in the wild cattle without horns would be at a serious disadvantage).
1. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4508108/
2. https://www.nature.com/articles/jhg201296
3. https://onlinelibrary.wiley.com/doi/abs/10.1002/ajmg.a.60693 376