The constant back and forth from Gotham to Metropolis in BvS is downright confusing because they make no effort to differentiate them, it's impossible to tell where a scene is set.
Since the early days of DC Comics, both Metropolis and Gotham City were presumed to be based on/inspired by NYC. Indeed, “Gotham” is one of NYC’s many nicknames. And “Metropolis” typically refers to the biggest, most important city in a region or country — a clear allusion to NYC.
But, fictionally speaking, Metropolis and Gotham were obviously not the
same megacity. So by fanciful geography, they were each
separately located “somewhere” on the US northeast coast. (Whether NYC
also exists within the fictional geography seems to depend on the story/continuity.)
In terms of distance between the two, several iterations (long predating
BvS) depict Metropolis and Gotham as “twin cities” separated by a modest body of water. According to some references, this resembled the Narrows between Staten Island and Brooklyn. According to others, it was Delaware Bay (which would situate Metropolis in Delaware and Gotham in New Jersey).
With respect to
BvS in particular, it’s pretty clear that this “twin cities” concept was being employed. So it was of little practical/plot relevance where,
specifically, various scenes were taking place. It was all within the same “megalopolis.” And most of the time, the location
was obvious— be it Metropolis proper, Gotham proper or somewhere in between (e.g., Stryker’s Island).