From what I've seen, the majority of fans think that Budapest had some huge romantic significance (or they did pre Age of Ultron). These are the main themes I've come across.

I think differently though: I just think it was a random mission, with some similarities to New York which is why Natasha makes the comparison, though obviously lacking in aliens. I've come across one fan fiction in which Natasha doesn't remember Budapest due to a head injury and being drugged while captive, and she keeps throwing random guess references to try to get Clint to say what happened, then Budapest becomes an inside joke for them. I liked that idea too :)