In fairness, I think a lot of that is him writing in a vernacular English that wouldn't have been recorded much in official documents or other writings. So he probably didn't coin it so much as save it for posterity. Otherwise his plays would have been like those fantasy novels whether the author gets slightly too excited about using their made up words for everything.
I agree. There are plenty of words for which the earliest surviving example of it being used is in a work attributed to Shakespeare, but there is no reason to believe that Shakespeare was the first to use any of those words. People don't generally use a new word for the first time in a play or a poem, but published plays and poems is all we have by Shakespeare.
The first use of a new word tends to be in conversation (so the first use is not recorded) or in a non-fiction work, where the meaning of the new word can be explained. Or in fantasy novels, as MrJohz points out. (But what's the earliest example of the sort of fantasy novel that has invented words in it?)
Some (many?) words attributed to Shakespeare follow the ordinary processes for making new words in certain open categories in English, and so you could argue that anybody might make these words as a competent English user, grasping for this specific meaning, has available the tools to make this word for that meaning. This could happen completely independently, even many times. If you need "invulnerable" or "majestic" for example you don't require the services of some specialist word inventor to come up with them, you can just make those yourself since you're a competent user and there are rules.
But Shakespeare used them in a context which people remembered, so that's why it matters, and in the process Shakespeare gets to choose some parameters for these words which are not set by the rules such as their emotional valence - is this a good thing or a bad thing? The rules usually don't dictate that, but usage does and Shakespeare can set the trend.
