Prof. Greenblatt contends in one place that,
Shakespeare, who had an indifferent or ambivalent relationship to print, seems to have cultivated a certain anonymity.
But I know not 'seems' and literary studies are lucky to be in a position where what seems rather than what is will suffice for a thesis. Of course, science is also only theory but it is in pursuit of laws, constants and truths rather than content to accept what seems as a conclusion.
It is frustrating for biographers of Shakespeare to find their subject quite so elusive and having to use so much of their own fancy to bring him into focus but it doesn't follow from that that Shakespeare's shadowy existence, from our perspective 400 years later, was a deliberate act of subterfuge on his part. That his age put less store by biographical detail was not his doing.
By all means, poets like Francois Villon or Ovid told us plenty about themselves in their work and Greenblatt's point is that Montaigne himself was the central figure in his own work as an essayist. But those were poets and an essayist whereas Shakepeare was primarily a dramatist, working in a professional theatre producing plays, sometimes in collaboration with others, plays that were owned by the theatre rather then their authors and, having been paid, they had less concern about the cult of the creative artist, the primacy of the text or even, perhaps, posterity.
Shakespeare apparently oversaw the first edition of the Sonnets, which had his name on it, and his plays were the most commercially successful of his day which, combined with being an actor and shareholder in the theatre company, made him a wealthy man and so he was well-known enough. There is no suggestion that he avoided celebrity status just as there is none that he pursued it. Even the most assiduous of scholars find it hard to establish a personal statement in Shakespeare's plays or poetry although James Shapiro's chapter on Hamlet in his book, 1599, makes a convincing case for that being his most finely-worked and thorough-going essay on a world view.
There was no attempt to hide authorship of the poems, The Rape of Lucrece or Venus and Adonis, in their dedication to a nobleman and the will includes bequests to friends and those he had worked with, it is thought, in his own memory.
But it is going at least one step too far too deduce a longing for anonymity in Shakespeare and it is a regular fault of modern biographers, if not all of them, to read more into the extant evidence than we have any right to. The most common tendencies are to make Shakespeare into a version of oneself or assume eveything good about him and refuse to entertain anything detrimental. The fact that Shakespeare died 200 years before the Romantic age that brought the personality of the poet to the centre of the work should always be there to remind us that we need to try to understand him from a pre-Romantic point of view. We don't seem to be very good at that.