I've seen this in a bunch of movies ranging from super serious period dramas like Tinker Tailor Soldier Spy to blockbusters like The Avengers where black widow mentions some past operation in Budapest. Why does Budapest keep showing up in this context?
Hungary occupied a slightly ambiguous place in the politics of the Cold War. Like Czechoslovakia, Hungary tried several times, unsuccessfully, to leave the sphere of Soviet influence.
Budapest was a significantly more open city than most of the other Eastern-Bloc capitals, and it was possible from a very early date for Austrians and other neutral nationals to travel to Budapest on a tourist visa.
At the same time, the goulash communism system implmenented in 1962 meant that starting the late 1960s, Hungary enjoyed a higher standard of living than most of the other communist bloc nations, and consumer goods were somewhat easier to come by. This made Hungary a popular tourist destination for people in other Eastern-Bloc nations, including the Soviet Union.
Thus, in the later cold war, Budapest was a place in which is was both possible and plausible for a Soviet citizen and an British citizen to both visit the same city and have lunch in the same coffee-shop or cafeteria. You can see why this would be a useful scenario for both spymasters and writers of popular fiction.
Source: The Reconstruction of Nations
One of the reasons has to do with the practicalities of filming. Budapest has a lot of different areas / sections, with different architectural styles, and therefore it can be made to look like a lot of different European cities where it might be more expensive or more cumbersome to film. It has been made to look like Buenos Aries, Moscow, Paris, Prague, Vienna...