I don't think a City Goblin is going to be viewed as being truly a Goblin by other Goblins, but Goblins that live in ghettos in Human cities are definitely not going to be seen as being Human either. It sort of comes down to how different immigrant communities in different places are.This seems unlikely, unless the goblin is fully integrated into this city's society. More likely, the goblins in a city are going to congregate in a specific section and maintain elements of their culture there. Goblin markets, goblin gambling dens, goblin music, all that stuff would be there, and the goblins maintain whatever gender norms they have there.
I can definitely picture a PC with a City Goblin background, really having problems talking to other Goblins because they're hanging around the 2 humans, a Dwarf and a Tiefling that's in their party.