The first thing to acknowledge when observing that there is an impersonality to the art of Shakespeare's characterizations is that you could and should say this about any good dramatist.  A work of drama differs from a novel, and a major way in which it differs is how the story is told.

In a novel, there is always a narrative point of view -- first person, second personal, third person -- through which the story is told.  This makes the work feel "personal," especially if it is written in the first person.  Those reading a novel critically often look for the point of view to give them information about the writer's own point of view, sometimes noticing that this or that character seems to "represent" the author's viewpoint.

In a play, the author must create all characters with equal fullness.  There is, to be sure, a protagonist, but, the play is not told from that character's point of view.  So, in drama, there is no narrative mode to give the reader (or audience member) a link to the author's personal perspective. Shakespeare was certainly a master of creating all his characters with equal fullness.

Another reason that we should regard Shakespeare as an impersonal writer is the simple fact that we know so few real facts about his life.  There has been tons of speculation, in fact, that these plays were written by someone else altogether.  It is hard to be very analytical about the relationship between the author's personal point of view and the characterizations in his work if we can't even agree who the author, in fact, is.

So, whether you are considering that plays always present an impersonal point of view, or you note the few facts (and authorship controversy) about his life, Shakespeare's plays must be noted as impersonal in their characterizations.