Out-of-universe, I suspect they hadn't decided yet that the Tesseract or Loki's staff held an Infinity Stone. That fact raises a number of questions, not the least of which is "Why was Thanos willing to give away one of the Infinity Stones he's so desperate to get?". So at the time the dialogue was written, it's most likely that the idea was that Loki's staff was powered by the cube in some sense. We don't even hear the word "Infinity Stone" used on-screen until the end of Thor 2 a year later, and even then it's basically just an Easter Egg.
In-universe, Fury was just wrong. His conclusion makes sense based on just what we see in Avengers, so it's not surprising he made the mistake. The connection between the two objects remains all the way through to the end, where we see that the scepter has the power to close the cube's portals.
Whether or not there is any kind of link between the Stones, we don't know. Since the end of Avengers I don't believe we've seen two of the Stones in the same place at the same time; the closest we've come is that we know The Collector had two of them (the Reality and Power Stones) in Guardians of the Galaxy, but the Aether was never shown or mentioned on screen.
It's possible that in Infinity War we do find out the Stones can interact with each other, but until then, all we know is that Fury was mistaken.