It's unlikely that Shakespeare ever had the opportunity to sign his work. His plays weren't even gathered and bound until after his death, and like today, people in his time did not walk around reading plays in their spare time.
But I recently described my version of an author talk, and it includes recommending books that I love, particularly if that book has a story or personal anecdote associated with it.
I try to be a storyteller at all times.
I also explained that if I'm in a bookstore for my talk and someone purchases on of these recommended titles, I'm more than willing to sign it, either with my name or , even better, the author's name.
I'm not sure if this would upset authors, but I'm promoting their work and bringing them sales. I can't imagine that they would complain.
Shakespeare would absolutely not complain, since he is dead. And since I often speak about Shakespeare during my author talks (I teach Shakespeare to my fifth graders and they perform a play at the end of the year, so have many amusing stories about the process), I've signed my share of his plays in my time, including last weekend, when I had the pleasure of signing a copy of King Lear at a book retreat in Vermont.
It was almost more fun than signing my own books.