There is, in fact, an instance of a single city split across two jurisdictions, albeit not an international border. That city is Lloydminster AB/SK. It’s a single city with a single government. The entire city is considered part of both provinces.
North Kansas City, MO is a municipal enclave, completely surrounded by, but not a part of, Kansas City, MO.
Since borders have come up, there’s one instance where the cities are in three states: Sioux City IA, North Sioux City SD, South Sioux City NE.
I meant the case when there are, politically, two cities, with two different mayors, but you go there and it is a single agglomeration with the border (possibly an international border) just running along the the middle of the sidewalk, right through a building, etc. There are not two disjunct urban centers. In that sort of situation, the two cities are more likely to have the same name, or pretty close, rather than North/South/East/West X. (Not sure how officially official East/West Berlin was as a name.)
If it’s really one city, as in @dtlique 's example, or e.g. the village of Ghajar in Israel/Lebanon (that one is no city, though, just a little village caught straddling the border so it’s not precisely the same type of thing) then obviously you would not expect multiple divergent official names.
If you want really weird (that word has been coming up a lot lately) situation of a town on a border, you can’t beat Baarle in the Netherlands/Blegium. The two parts are known as Baarle-Hertog (Belgium) and Baarle-Nassau (Netherlands).
On a recent trip through Pennsylvania, I spotted the town name “North East”, and was reminded of the town of the same name in Maryland.
it’s a pity there are no suburbs of either. East North East, or maybe North North East, or whatever.
Neither has Newburgh Heights, Ohio, a suburb of Cleveland. Also in Cuyahoga County, the city of South Euclid is south of the city of Euclid.