This myth appears to stem from The Talmud - a vast collection of Jewish laws and traditions.
Although modern-day medicine and science may beg to differ, the sages of the Talmud were under the impression that eating fish and meat together is extremely dangerous to one’s health. In fact, in those days, there was actually a dermatological condition which was believed to be caused by eating fish and meat together. As such, the rabbis prohibited the consumption of such mixtures, a practice which continues to this day. This ban applies to fowl as well.
Clearly, however, today this is not the case. Some try to reconcile this medical contradiction, claiming that what was unhealthy in those days may not be in ours. In fact, in talmudic times it was believed that rotten fish was good for you!
It is important to note that this prohibition is unlike the one forbidding mixing milk and meat. Here, it is merely forbidden to eat fish and meat at the exact same time, or in immediate succession. A waiting period, however, is not necessary. The dishes used for eating them may be interchanged, and hence no need for special “fish dishes” in one’s kitchen. (Explained by Rabbi Ari Enkin )
The skin disease particulars you mention above seems to be an elaboration from Hindu religion:
In Pakistan, it is widely believed that if one drinks milk after eating fish (or other way around), you get skin disease. However this is not true. The skin disease (in which people's skin become white overtime) is a tropical skin disease and has nothing to do with simultaneous consumption of fish and milk. In fact, this is a Hindu religion inspired idea and that is why it has become such a powerful belief in Pakistan (due to our pre-independence days) 
A 1996 published African article titled
« After Fish, Milk Do not Wish ». Recurring Ideas in a Global Culture , did a thorough investigation of the origin and evolution of this, and related myths, as well as referencing a handful of scientists that performed more scientifically thorough investigations:
A widespread belief in Mali associates certain forms of fish diets with leprosy (Hansen's disease). Very similar aversions to fish
diets have existed in ancient Egypt, the Islamic Middle East, northern Nigeria, medieval Europe, and colonial America.
The beliefs documented by Robinson were listed as Hausa 'superstitions and customs' in 1913 monograph Seven years earlier however respected and well-known British leprosy
specialist Dr Jonathan Hutchinson promoted virtually the same beliefs as
scientific hypothesis in 407-page book, On Leprosy and Fish-Eating
Statement of Facts and Explanations (1906) Synthesizing data collected
from around the world by himself and by others Hutchinson argued
vehemently against the then prevailing contagion theories He suggested
instead that the 'cause of the disease is some ingredient or parasite
generated by or introduced into fish which has been either not cured at all
or cured badly.'
As late as the mid-1960s handful of doctors trained in scientific
medicine were still earnestly investigating the fish hypothesis Meny
Bergel director of the Leprosy Research Laboratory in Rosario Argentina
published several articles 1958195919601966 attempting to explain the
well established correlation between the occurrence of leprosy and the
ingestion of diets high in decomposing fish In short he found that the
consumption of rancid fats and unsaturated fatty acids favored the growth
of the Hansen bacillus in laboratory rats Though never harboring the
actual disease fish nonetheless facilitated its development.
Their references only reach as far back as 1995. I am doubtful that any scientist would have more recently dignified this claim with an investigation. Any individual doctor or nutritionist, when consulted, is able to debunk these fears with basic facts of medicine and chemistry. Over time, the nature of the claim has changed (from causing death, to Leprosy, to Vitiligo, to generic skin disease, to generic bad health - but mostly Leprosy). There's zero evidence for these claims in modern times. The cause for Leprosy is now well known and has nothing to do with fish or milk or any combination of the two.