Why is the title of the movie The Karate Kid (2010) when all the characters are using Kung fu?
I even remember Dre, at some point, mentioning karate just to be corrected that it is Kung fu.
Because it's a remake of a film with the same name and they didn't want to lose out on the popularity of the older film. If you have seen both films, you will notice that both stories are quite similar too, only replacing Karate with Kung-fu and changing the location.
Sony, at one point, thought about changing it but they didn't in the end:
Despite maintaining the original title, the 2010 remake does not feature karate, which is from Okinawa, but focuses on the main character learning kung fu in China. Chan told interviewers that film cast members generally referred to the film as The Kung Fu Kid, and he believed the film would only be called The Karate Kid in America, and The Kung Fu Kid in China.
Sony had considered changing title of the film, but Jerry Weintraub, one of the producers, rejected the idea. Weintraub was also the producer of the original Karate Kid. - Wikipedia
The 2010 movie The Karate Kid is a remake of a 1984 movie by the same name. In that movie, Daniel learns Karate and participates in a tournament.
Miyagi proposes that Daniel will enter the Under-18 All-Valley Karate Tournament, where he can compete with all the Cobra Kai students, and he requests that the bullying should cease while Daniel trains. Kreese agrees to the terms, and warns that if Daniel does not appear at the tournament, the harassment will resume on Daniel and Miyagi.
Miyagi begins Daniel's training by telling him that Daniel is required to commit unequivocally to Miyagi's curriculum, and to obey Miyagi's every instruction, without question. After Daniel agrees, Miyagi assigns Daniel to complete various lengthy, menial chores that appear to have nothing to do with karate.