While watching some videos/movies or reading books in English, I tend to see that people always adding the word "city" to New York (New York City).
What's behind this stuff in English?
Adding the word "city" is not "stuff in English".
The name of the city is New York City. Quite often, though – perhaps because it happens to be one of the biggest and most famous cities in the world – this is shortened to New York.
To make matters a little bit more confusing, New York City happens to be in the state of New York, so, when someone says, "I drove through New York last month," that could mean a person drove across the state, or it could mean they drove through the city. But all ambiguity could be eliminated if they say, "I drove through New York City last month," or "I drove through New York state last month."
This is more a quirk of geographic names than it is a quirk of English. There is a town called Nebraska City in the state of Nebraska, but no one shortens that to Nebraska. When they say "Nebraska," they mean the state, and when they refer to the city, they call it "Nebraska City." The same is true for Oklahoma City, Oklahoma. I have never heard anyone from Traverse City, Michigan refer to the city as "Traverse" (although I have heard it affectionately called "TC" by locals.) And, for the most part, no one calls Seattle, "Seattle City", or refers to Miami as "Miami City" (although you might read or hear "the city of Seattle" from time to time).