Yes, this is how I see Fury and Barton too... like he basically put Barton in charge of overseeing the tesseract, which was obviously a pretty major thing. I haven't read the comics but I read some stuff online (which of course may or may not have validity) early on about the Ultimates (I think JR mentioned the Hawkeye in the films is based more on the Ultimate Avengers) where Hawkeye and Fury had that sort of relationship. Of course in the films, Natasha also seems "close" to Fury, he put his arm around her briefly in IM2 and she and him were arguing with each other in the Avengers (of course that's with the spear making every agitated). But yeah, my impression is that Clint is pretty high up and both men respect each other...even more evidenced in the Avengers deleted scenes (Clint "missing" shooting Fury, Fury trying his damnedest to protect and get Clint back).

I will have to rewatch the scene with Natasha in the Avengers. I have only ever assumed he was slightly making fun of her (like the Budapest line), because of his delivery.