The education Shakespeare received would not have encompassed the themes and references visible in his work.
Shakespeare’s plays show an author well-versed in mythology, law, astronomy, music, foreign policy, the military, politics, foreign languages, the royal court, European geography and history. But Shakespeare was not educated past the age of 13.
Shakespeare's work displays a broad vocabulary and draws on a wide range of source material, some of which was only available in foreign languages. His curriculum at the local grammar school in Stratford would not have exposed him to these themes or works. The standard Elizabethan curriculum at the time would have consisted of rote learning in Latin. The school would likely have taught classes in arithmetic, logic and rhetoric, with no hint of law, history, medicine, art, military strategy, or European language.
There is no reason to believe that Shakespeare did not travel. He may have been in the army for a period and he could have been a tutor. It is not inconceivable to believe that he did, over the course of his life, come to know enough of all of these subjects to create the material present in his plays. Also, Shakespeare was educated at Stratford Grammar School. He would have been exposed to a curriculum of Greek and Latin and could have featured foreign languages. The actual Latin book used in the school’s syllabus was actually mentioned in the Taming of the Shrew. Through his work with a royal acting company, Shakespeare would also have likely been exposed to the workings of the aristocracy and courts. Also, many of the best writers of Elizabethan England did not come from noble stock. Ben Jonson was the son of a bricklayer. Edmund Spenser's father was a tailor. Christopher Marlowe was the son of a butcher. The idea that only someone with access to a noble education could produce great literary fiction is utterly ridiculous.
[P1] The education Shakespeare received at Stratford Grammar School did not expose him to the source material and themes used in his plays. [P2] Therefore, the boy from Stratford is not the author of Shakespeare's works.
[Rejecting P1] His education would have touched on many Latin and Greek themes visible in his plays. [Rejecting P2] Just because he didn't learn about politics, royal court, history, law and military strategy at school doesn't mean he never learned it.