There are some practical ways he has influenced culture: firstly, he was writing at a time when language was expanding. He coined hundreds of words that have drifted into everyday speech, so many that we entirely forget they were invented by Shakespeare. Fundamentally, he was a soundbite person, and that profoundly altered the way in which we communicate, even today.
Secondly, his current influence is also seen through his inclusion in the education system, often as a mandatory feature of literature courses. That’s why there are as many people in the UK who hate Shakespeare as love him, largely because of their experiences at school. It’s a curious change: Shakespeare was a playwright who was doing exciting, daring productions, but he has become somewhat churchy and a sign that you’re intellectual if you can quote him. He has become something very other from what he was.
As such, he has affected culture in ceasing to be himself: he was a local playwright writing for London, whereas now he has become a global figure reinterpreted in in countless new contexts. His malleability is integral to his ongoing influence.