It seems to me that the way neural networks are trained is similar to the way we "train" or educate a child (or a person, in general).
Can an AI eventually think like a human?
In theory, yes. You need to truly understand human mind before you can say that machine mind should exactly work the same way. However, the challenge of understanding human brain is probably harder than building a machine that seems like human in every way from outside but may not think the same way.
Instead of building machines who can play Go and recognize handwritten characters, the next step in AI would be the development of systems which can think like humans and learn like humans. But what does that mean in reality? It means, to use techniques like qualitative physics engines, grounded natural language, modelbased bootstrapping and story based intelligence. If a cognitive system contains all of them, it's inner working will be much more human-like than today's AI software. Mastering this task is not easy, because nobody can define exactly what story based intelligence is. But there is a shortcut available. Instead of programming software which can make decisions, the easier step to implement is writing programs which can recognize what humans are doing. It is possible with little amount of effort to realize systems which can parse natural language, parse a story plot and recognize if something works according to a qualitative physics engine. Or to say it shorter: to detect sense.
And yes, the resulting sourcecode will behave more human-like than the examples from the introduction. Because what humans are doing is model building, creating ontologies, using natural language and investigating story plots. To realize such capabilities in software it is important to focus not on machine intelligence, but onto human intelligence. That means, if somebody is an expert for English language, he is also an expert for english language which is parsed by a context-free grammar.