Sorry for double posting, but I forgot to point out a problem in "chaining modifying clause".
Actually, my example 鱗が小さい魚を食べる猫が嫌いな人 can have another meaning; a person who eats a fish whose scales are small, and (who) dislikes a cat. In this interpretation, the clause 鱗が小さい魚を食べる modifies 人, not 猫. A comma is often inserted after 食べる in this case (鱗が小さい魚を食べる、猫が嫌いな人), and the meaning is often obvious from the context, but you need to pay attention to this problem when making or reading these types of structure. In fact, the same goes to your example 魚が好きな人が好きな人. This can mean "a person who likes fish and people" (魚が好きな、人が好きな人). Other expressions 魚と人が好きな人 or 魚が好きで、人も好きな人 are far more commonly used for this meaning, though.