It's down to what the chicken ate while it was alive. Saturated fat sets, olive oil sets if you chill it, but not otherwise, and a number of seed oils do not set (rapeseed for instance). When you make a stock which has solidified fat on top, that's saturated fat, so I'd hazard a guess that the stock where the fat doesn't set means a healthier eating chicken, because it contained less saturated fat to start with.
UPDATE: Thank you to the person who bothered to do the research and said my answer 'might have some merit'. Chickens are no different from human beings - the fats you put in are the fats floating round your bloodstream and depositing in various places; think about corn fed chickens, where the fat composition is slightly different, not to mention the colour of the flesh itself. That will be a partial explanation; when taken together with the fact that not all chickens, even in the same flock or brood, get to eat the same diet, because the pecking order dictates that some free range birds don't always get the pick of the food, explains differences in chickens from the same supplier. Of course, if you can come up with another explanation, I'd be delighted to hear it...
UPDATE 2: Perhaps I should have been clearer. I am not for a moment suggesting that the fats eaten are deposited in their original form, but if you know anything about biology (chickens or otherwise) then you'll know that certain synergies occur, depending what's put in, which change the composition of any fats deposited within the body system. Hence the connection between eating lots of saturated fat and having high cholesterol in humans, for example.
UPDATE 3: Rumtscho: Can't find any scientific evidence so far to prove this theory regarding chickens, but, for interest's sake, and to prove how much of a difference it can make, farmed salmon in Britain no longer has a balanced omega 3/6/9 ratio, as it should do, and still does in the wild. It's because the feed had to be changed, and the consequence of that has been a much higher level of omega 6 in particular. I'm still looking for something on chicken.
UPDATE 4: Now I've had time to look properly, it's not at all difficult to find scientific evidence, there's plenty of it. There's a study carried out by The American Society for Nutritional Science in 2000 comparing the fat deposition (and other metabolic processes) between chickens fed the same diet, but one lot with saturated fat included in the form of tallow, and the other lot with polyunsaturated fats. The fat deposition in the birds fed tallow was greater, and the composition of the fat contained more saturates compared to the polyunsaturated group. These results reflect previous studies (Sanz et al 1999 and 2000).
Effectively, it's like everything else - you get back what you put in.
Best Answer
I'm not sure what might have separated in your stock, but I do know that 8-9 hours is way too long for a fish stock.
While its common to simmer beef, chicken, and other meat stocks for that length of time, it's much too long for fish stock. Because fish bones are so thin, they cook and soften much quicker than harder bones from land animals. This means that a delicious and flavorful fish stock can be made in as little as half an hour and rarely takes much more than that. Overcooking fish bones can lead to a bitter and unpleasant flavor, so your stock was likely ruined to begin with even before it separated.
The leathery bits you describe are likely proteins released by the fish parts used and re-coagulated by extended cooking time. I would guess that the oily part is at least full of fat/fish oils, if not completely made up of it-- you don't say how much you have, so I suppose it is possible that if you've been simmering it all day you've managed to cook all the water out. If that's the case, then you may have accidentally made fish oil, which many people take in capsule form as a health supplement.