Sure, playing chess needs intelligence, dedication, and good chess players are smarter than an average person. But it’s waaaay exaggerated in movies. I’m a math researcher, and in any movie, my department will be full of chess geniuses. But in reality, only about 10% of them even play chess.
Good chess players, though, exhibit some common traits which are shared with “smart people”: the ability to think in abstract terms, and a good memory.
Your success at chess is often based on how far in advance you can plan a game at any point on the board, greatly supplemented by your ability to remember entire games of famous matches. These skills are frequently exhibited by people considered smart. However, as you and OP point out, you have to play, practice, and memorize to get good; merely knowing the rules and being smart doesn’t get you there.