The core problem I see with Inheritance is that the abstractions tend to fall apart and no longer be true. Lets use the Animal example. It is easy, when you have Animal -> Cat and Animal -> Dog. But as soon as we become more specific like Animal -> Mammal -> Cat, Animal -> Fish -> Hammerhead Shark, Animal -> Bird -> Bald eagle, we risk of getting in trouble. Because now for all purposes we assume things about the Fish, Birds and Mammals, like fish is in the sea and mammals are live on land. We know that this is not strictly true, but for everything we need it works. Later we need to handle a dolphin… should that be a fish, or do we need to restructure the whole program. If we treat it like a fish, then we might be even deeper in trouble later if we would need to handle birth. And even if we restructure our program to be correct to handle birth, we might stil forget that some mammals lay eggs like the Platypus, so then things break again if we need to handle that. We tend to see Inheritance as a rigid fact based structure, but the core problem is that it is just a bunch of assumptions we dictate in a very rigid way that is hard to change.
Composition have no problem with specifying the platypus as a mammal that lays eggs and have a beak.