This is a great question because over the decades people have expressed a wide range of opinions about the "relevance," or lack thereof, of The Catcher in the Rye. As early as the 1970's, some people regarded the novel as dated. There were seismic changes in American society during that time, making the pushback understandable. But over the decades, more people have thought it was relevant than out of touch. Its defenders believe that the book's classic status is due to its expression of the universal angst of youth as it is in all times and places.
Still, even if the novel is timeless in its way, most people would agree that the story as written could not take place in the year 2020. It could not have taken place even several decades ago. For one thing, Holden is far more naive than any teenager is likely to be in our time. His attitudes about women and sex, for example, seem more like those of an eleven- or twelve-year-old than of a young man Holden's age.
Even if his thinking and his reactions to people around him might somehow still be realistic in the twenty-first century, the principal element in the book that would have to be changed is its language. Both the dialogue and, perhaps even more so, Holden's first-person narration would be radically altered if the story was placed in 2020. The whole style of the novel is filled with expressions and mannerisms that are no longer used today. Phrases like "strictly for the birds" and "give her the time" are not in common use. Also, demeaning references to gay people as "flits," and even the repeated use of "goddamn" instead of the f-word, are extremely dated.
Apart from this, the story does not fit into our present world because unusual things in it would no longer be seen as strange way now. For example, a high-schooler having a prostitute come to his room in 2020 may be viewed in some circles as a kind of curiosity rather than the (to put it mildly) risqué thing it was in the 1950's.
The question then remains as to why, if The Catcher in the Rye would have to be different if written today, it's still so widely read and so moving as it is. Part of the answer is that the novel is interesting precisely because it shows a world different from ours. Even if a book is "dated" and could not be written the same way now, its central meaning may still be universal and timeless, despite the trappings of language and circumstances that mark it as a work of the past.
Holden's situation, that of a lonely, angry and isolated young man, is a phenomenon that has existed among people in all times and places. He's a sort of 1950's version of Hamlet, speaking universal truths and feeling emotions that everybody feels. The reader doesn't have to have lived in Holden's own time to understand him, just as we don't have to be a medieval Danish person or an Elizabethan in order to grasp the truths that animate Hamlet.