I'm not sure I understand your question, but a big part of the reason no one today writes like Shakespeare is because Shakespeare was, without doubt, the greatest English language writer who ever lived, and no one, anywhere, in any time period, can come close to writing as well as he did.
And if you mean the kind of stories he told, writers have been copying them since before he died. His stories have been rewritten thousands of time, told a thousand ways, and will be until we all shuffle off this mortal coil.
Did he take ideas from other sources, and rewrite older stories? Yep, and so do we all. Anyone who thinks otherwise probably believes he is Shakespeare, in fact, and may, or may not, be let out of his straitjacket someday.
It ain't the idea that matters, it's how well you write it. Shakespeare write them better than anyone else ever has, or ever will.
Though anyone who thinks all of Shakespeare's plays were taken from other writers has read very little Shakespeare, and no history. Many of his ideas and plays were original, and where and how he found the information to write them remains a mystery.
If Shakespeare were alive today, he'd be writing in modern language, but his talent level would still be higher than any other writer, his genius would still be higher, and he'd still be the greatest English language writer of our time, or any other.
But, in short, no one today writes like Shakespeare because no one today is capable of writing like Shakespeare.
May I say, the Kenneth Branagh adaptation of Much Ado About Nothing was a big hit in 'modern' times... watched a 'rerun' recently and it was so funny (ETA: of course, I am predisposed to like it perhaps
