One cause of the perception is a conflation of novel and creative uses of existing words and neologisms. Shakespeare was clearly highly inventive in how he used existing words in new senses.
Also, there is conflation between words and phrases. Phrases like “one fell swoop” and “sea change” were probably first used by the Bard (the latter almost certainly so). But that’s not the same thing as coining a new word. It’s not that Shakespeare was “coining phrases,” but rather the phrases became ingrained in the lexicon to the point of cliche through endless quotation.
But while he was creative in the use of language, there is nothing superhuman about what he did. Some other writers were just as creative, only we don’t notice it because they’re not Shakespeare.