As Bill Bryson Notes in his edition of the Eminent Lives series, the history plays were highly reflective of the mood and political movements of the time and always looking to curry favour with the Monarch as well as provide entertainment for Londoners of all classes. This, it is believed, is where we get the warped interpretation of King Richard the 3rd as a person and ruler - as Shakespeare was almost demonising the last Plantagenet King who was superseded through war by the Tudor Dynasty, to which Elizabeth belonged.

This point would naturally provide inaccuracies in terms of character and actual legacy of the historical figures - but also, there are some hard-factual inaccuracies too. For example, as Bryson puts it,

"Whether by design or or from ignorance, he could be
breathtakingly casual with facts...In Henry VI part I, for example, he dispatches Lord Talbot twenty-two years early, conveniently allowing him to predecease Joan of Arc.

Also, again either through ignorance or poetic license, there are Geographical and seemingly obvious inaccuracies in his comedies and tragedies. A glaring example is,

"...In the Tempest and Two Gentlemen of Verona he has Prospero and Valentine set sail from, respectively, Milan and Verona, even though both cities are a good two days' travel from salt water.

So although Shakespeare it is believed had a good education in Stratford, his plays [history, comedy and tragedy alike] abound with factual errors, embellishments and misplacements. Whether this was through ignorance, poetic license, fear of arrest for outraging the nobility/monarch is a matter of speculation. His more worldly audience (such as merchants and the nobility) would have known to a certain extent, whether the common audience would have cared is unknown, but royal and aristocratic patrons definitely would have cared if the history plays showed them and their ilk in a less than positive light.

Edit: At present, my only quotable source is this book by Bill Bryson, as I have literally just finished reading it and am away from my bookcase. I will add further citations from other works if people request, but it won't be for a good few days!