I have totally different types of coats between my two blenheims, between the two tris, and between the blenheims and the tris. The type of coat the parents and grandparents have is probably the biggest determinant for coat, I think (think about our own heads of hair and how one can have mousy limp hair and someone else, a full head of think curly hair! It ain't fair but it is the distribution of the genes... :lol
.-- unless an animal is fed a really poor diet I don't think supplements or particular diets make much difference to coat appearance. I think people tend to see what they except to see as a result of giving the type of diet or supplements.
It's simply a variation of the very well known placebo effect -- people tend to see or feel the improvement they expect to see; it's why double blind tests are so important for gauging the actual effect of a diet, a drug, a supplement.
I've had rescue dogs come in with lovely coats (eg Parker, who went to Sins' parents) that come from backgrounds where they were fed absolute crap in the former home, so diet as long as it meets standard requirements, simply cannot make too big a difference I think. Manwhile Lily (tri), who gets a great diet here, has a so-so coat, while Leo (tri) is nice and glossy.
There are some variations by colour though -- black hair tends to get fullest and longest and I find -- sheds the most.
A lot of coat appearance can come from the type of conditioner used by the groomers or in the home. Conditioners or spray-on grooming aids with silicone impart a really nice sheen and silkiness to the coat. These can last for weeks.