1. Both were collaborators. Shakespeare wrote plays. They were meant to be seen on stage, not read as scripts, as they are in so many English classes. You get the full flavour when actors bring them to life. Alan Moore writes comics. They're illustrated by pencillers, inkers and colourists. You can find a few of his scripts if you hunt them down, but that's not the way you're supposed to experience his stuff. The pictures complete the… picture.
2. Both love/d language. Their dialogue teems with puns, references, alliteration and poetic flair, all of which still works to tell a story that grabs you by the balls.
3. Both mixed low art with high. Sex and violence and intrigue and pratfalls alternate with penetrating insight into the human soul. Both understood/stand that life contains all of these things in an inseparable multi-coloured swirl.
4. Both were/are prolific to a degree that would make most writers weep. Gaze upon the thickness of their respective catalogues and despair. And all of this work was/is done in combination with the talents, schedules and contributions of their collaborators.
5. Both took established characters and did their own thing with them. Shakespeare took big liberties with the historical figures he wrote about (in actual life, Macbeth ruled Scotland for seventeen years, and Richard III was a nice guy). He did his version of Julius Caesar. The Comedy of Errors was a rewrite of two plays by Plautus. All's Well that Ends Well is the same story as one of the tales in The Decameron. As You Like It was based on Thomas Lodge's novel Rosalynde. He was an adapter, not an originator. Alan Moore took over Saga of the Swamp Thing in its twenty-first issue, kept the established continuity, and did something wholly new with it. From Hell tells the story of Jack the Ripper. The League of Extraordinary Gentlemen puts together various characters from Victorian fiction: Captain Nemo, Alan Quartermain, Mina Murray from Dracula, the Invisible Man, Mr. Hyde, and Moore takes them in his own direction. He penned multiple stories with some of DC's best and least known characters: Superman, Batman, Green Arrow, Vigilante, The Phantom Stranger. He wrote the Joker's origin.
6. Each wrote the work considered the pinnacle of his form: Hamlet and Watchmen.
7. Both were/are from central England. Shakespeare was born in Stratford-upon-Avon, worked in London, and retired back to Stratford. Alan Moore has only ever lived in Northampton.
8. Neither was/is recognized by the literary and cultural elite of his day. Shakespeare's epitaph makes no mention of this theatre career. Alan Moore just writes comics. But their fans adore/d them. And I wouldn't be surprised if just as many academics, hundreds of years from now, earn their bread dissecting Alan Moore's pages as Shakespeare's.
By the way, I do believe there's reasonable doubt that the man from Stratford wrote the works Shakespeare. But I'd lay good money on Alan Moore writing the works of Alan Moore.