Can I say, another example, 'the city London', without 'the', if I refer to the settlement itself, not "the city" within it (that financial district of London with skyscrapers and stuff)? I see no grammatical impediment not to do it. You wouldn't say 'the writer of Arthur Conan Doyle', right? Why wouldn't it work the same way with the names of cities?
Local idiom will define individual cases. So we have
New York City.
The City of London
This one refers only to the central financial district, within the old city walls and not the wider urban area, explicitly excluding Westminster, where Parliament, "Big Ben" and Buckingham Palace are located.
We don't generally use an appositive. We don't say "The City London". There is no logical reason for this. There is a poetic expression "London town" but this expression is idiomatic and you can't say "Paris town", for example.
Instead the most common is just to use the name of the city:
I'm going to London tomorrow.
I love New York.
The City of New York, usually referred to as either New York City (NYC) or simply New York (NY)... (source: Wikipedia)
"The city of New York" is the official name of the city. The prefix "The city of" is used in the official names of many cities in the United States and possibly in other countries. In most cases people refer to it as "New York City" or just "New York".
The reason that "city" is green attached after "New York" (more than other cities' names) is that New York is also a state, and in many cases there's a possible ambiguity. Similarly, the US capital is called "Washington, D.C." (and in casual speech usually just "DC" to distinguish it from the state of Washington (which is often called "Washington State").