From a chemistry point of view milk is an emulsion of cow fats dispersed in water (and some proteins, and electrolytes, not relevant to this discussion). Shaking up milk or using a churn breaks the emulsion and you get solid butter. Strangely about 95% of the population has no idea where butter comes from beyond "it comes from the store". Ordinarily water and oil don't mix very well. Some really tasty foods (not just milk) involve oil/water emulsions, sauces, gravys, salad dressings, mayonnaise, to a greater or lesser extent raw baked good doughs, etc.

So thats chemistry lesson 1, lesson 2 is "lots of stuff" in o-chem class prefers to partition itself in oil or water phase. So at least some ochem lab work involves a sep funnel which is a magic, yet simple, way to do liq vs liq extractions. Stuff moves from being dissolved in liq 1 to being dissolved in liq 2. Its a law of nature that its impossible for an undergrad lab involving a sep funnel to not have an accident or incident of some sort, usually more funny than dangerous. Not surprisingly one liquid is usually polar and one is usually non-polar because you can't do much of an extraction if liq 1 and liq 2 mix perfectly to create liq 3. And the sep fraction is often crazy, if you work the polar / nonpolar hard enough. I would imagine the sep fraction for salt or sugar in water/veg oil could be nearly a million, like the only salt in veg oil might be mechanical turbidity/suspension. Thats just a guess but I'm sure its ridiculously high if not a million. You can still have a "useful at lab scale" extraction if its only a factor of 2 or so, just pointing out its not shocking to be much higher.

Anyway the point of this chem discussion is there is no such thing as milk, there's a liquid that comes out of a cow and its got all kinds of crazy stuff in it, some of which STRONGLY partitions into either the polar or nonpolar phase. So the eventual discovery is probably going to be some kind of hormone, protein, mineral, herbicide, antiseptic, antibiotic, pesticide, solvent, "something" that strongly partitions / dissolves into fat and not into water. In the list above, hormones, herbicides, pesticides, and solvents tend to partition strongly into fat, the other stuff is a tough call one way or the other.

I would theorize that other animals living in the same contaminated conditions would partition the same way. Us mammals are not as biochemically different as some think. So its entirely possible you'd get the same effect from eating fatty meat, perhaps bacon, or foods cooked in (or made with) lard. Its worth some study.