From a sci-fi point of view, if anyone remembers the movie 'Enemy Mine', they both realize their 'holy books' are basically the same document, since we all love peace and such. So it's reasonable that Klingons might have had a famous playwright who wrote about death and kings and love and war; and, on reading Shakespeare might have said: "Hey! This is just like Ch'Tang (or whoever)" and so they might say he was 'the original Shakespeare' to humans (the alternative being that they were SO IMPRESSED with Shakes that they just tried to steal him, which seems unlikely).
I always thought that Chang was trying to one up Kirk and humanity within the context of making a joke. Like acknowledging that our Shakespeare is good, but that Klingons could've done him so much better.