The concept of the district likely stems from the Marshallian industrial districts, in terms of a highly concentrated and specialized area of industry, and then taking the concept and applying it to other areas of specialization in a way similar to that neatly outlined in Ed Glaeser's Triumph of the City.
So, in that sense, each district is far more than a building or a neighborhood. It's better to think of each district more as a full blown city in our sense, a specialized satellite city around the metropolitan center that is what Civ calls the city.
You can already see examples of the districts in early civilization, especially with the harbor. For example, Piraeus would be the harbor district of Athens, and Ostia for Rome. But it even extends to later eras and modern day, especially as cities become more spread out. Silicon Valley as the campus for San Francisco, Boulder for Denver, Oxford for London, Electronic City (Bangalore), Cyberabad (Hyderabad), and Pune (Mumbai) in India. For encampments you only need to look at military bases, and harbors, just as in ancient times, at navy yards (Brooklyn for New York, Alexandria for DC, etc.). Neighborhoods are basically just suburbs or exurbs. You can pretty easily find the same connections for other districts.