Fact
Shakespeare did not coin as many words as you were told
He is the first recorded use of many words. That is a different claim.
The figure of 1,700 invented words comes from counting first attestations in the Oxford English Dictionary. But the OED drew heavily on Shakespeare early on, and many supposed coinages have since been found in earlier texts as digitised archives expand. He was an inventive user of English; the invention count keeps shrinking.