Is Karate a Japanese sport

In short, yes.

Okinowan Martial Arts were eventually popularized by Gichin Funakoshi, who developed the Shotokan karate style. He took his style of martial arts on tour throughout the country of Japan, demonstrating them at colleges and other community hubs. Eventually many school adopted karate as a part of their physical fitness programs and many colleges have Karate clubs. It wasn't until after the second World War that karate spread out of Japan into western countries. By then it was "all the rage" throughout Japan.