Usually, 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? I can only guess that it's used just to distinguish state and City itself. However, in some other languages(e.g. Ukrainian), usually people don't use "City" after NY. They would assume that the person is talking about the city already. If you want to talk about state, you can mention that.
Your assumption is correct—it's to distinguish between the city and the state.
Idiomatically, it's simply the case that the phrase New York city (or the proper noun New York City) was picked as the more common "identifier" over than the phrase New York state (or the proper noun New York State).
(Having said that, I have heard people refer to New York State—but not as often.)
I can't tell you why this would have been the case originally, simply that it ended up that way and we continue to use it.
Update:
Per the Wikipedia entry for the history of the city of New York the timeline of its names is as follows:
1524: New Angoulême
1664: New York
1898: City of New York
And per the main entry for the city of New York:
The City of New York, often called New York City (NYC) or simply New York, is the most populous city in the United States.
Also per the Wikipedia entry for the state of New York:
1664: Province of New York
1776: New York
And:
To differentiate the state from the city with the same name, it is sometimes called New York State.
But, strictly speaking, the city is named City of New York and the state is named New York.
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).
If I tell people that my wife and her family are from New York, they frequently assume I'm talking about New York City (NYC). Only I'm not, I'm talking about a place that's 5 and half to six hours away by car. It's fairly important to say whether you mean NYC or New York State because the city comes to mind first for most Americans, not the state -- which is logical -- since 8 1/2 million people live there, making it the most populous city by far in the States. Far fewer people have been to what is commonly referred to by New York City inhabitants as "Upstate New York" or just "Upstate". This area is generally any area not New York City, as New York City itself sits on the southernmost latitude of the state in line with the state of Pennsylvania. Those people that do visit upstate generally only visit the Niagara Falls in Buffalo on the other side of the state, or visit the Adirondacks to the north.
