Shakespeare didn't write in Latin, but he did write Latin. His plays are full of Latin references -- and often jokes that are only funny to other people who took Latin in school. The Merry Wives of Windsor has an entire scene devoted to it, and even today it can be very funny. Monty Python didn't invent that joke -- and they sure knew their Shakespeare.

Shakespeare was less of a Latin showoff than Marlowe or Kyd. In fact he was mocked for his lack of a university education[1]. But he definitely considered Latin useful, throughout his career, and playwrights continued to reference it for many years.

[1] https://artsci.wustl.edu/ampersand/upstart-crow-shakespeares...