Alex didn't just associate a word with a specific object: he could generalize, which is usually something only humans can do. For example, when shown various keys, he could recognize that they were the same thing and even point out differences in size and color. If you still don't think that's impressive, take a look at this video of Alex interacting with his trainer, Dr. Irene Pepperberg, and try not to freak out when he asks her for some water:
Holy shit, that bird behaves more like a human than half the people we saw on the subway this morning. It's almost like he stepped out of a Pixar movie. One day, as Dr. Irene was closing up the lab, Alex said goodbye by telling her to "be good." Irene replied by saying "I love you," to which Alex answered, "I love you too." He then asked: "You'll be in tomorrow?" Irene assured him that, yes, he'd see her tomorrow.The next morning, Alex was dead.
There appears to be something in our eyes.
They actually had conversations like that all the time (you can see a similar one here, at around 7:55). You can say that he was just faking his banter and emotions through years and years of social repetition, but at what point can't you say the same thing about humans? You only know to jokingly tell someone to "be good" when they head out for the night because you heard somebody else say it and remembered it.