I believe what Bob is referring to by "Shakespeare's Shared Universe" (and oh sweet Christ just typing that makes me nauseated) is the history plays which are more than loosely connected as they cover the War of the Roses, commonly referred to as the First Tetralogy and Second Tetralogy (Tetralogy being the actual word for "series of four" because "quadrilogy" is not a fucking word). The First Tetralogy goes:
Henry VI Part 1
Henry VI Part 2
Henry VI Part 3
Richard III
Henry VI 1-3 is one of his earlier, less polished efforts, is incredibly long and boring, and is thus rarely performed. Richard III, of course, is a far better known and better regarded play, and one of the plays where Shakespeare seems to have matured as a writer. (Again, might have this wrong, so if you care, please correct me.) The Second Tetralogy, which is actually (urp) a prequel, goes thus:
Richard II
Henry IV Part 1
Henry IV Part 2
Henry V
Universally the Second Tetralogy is considered superior, mostly because Henry VI is an unbearable mess while all four of these are among Shakespeare's best. And as for recurring characters, yes Falstaff is one (and we'll get back to him in a second) but so is Prince Hal of Henry IV, who is, yes, Henry V himself in the eponymous play.
Now if Bob is talking about Falstaff as being indicative of a "shared universe" (blech), well, the Falstaff that shows up in The Merry Wives of Windsor really isn't very much like the one from the Henry plays, supposedly because the character was so popular Queen Elizabeth requested Bad Bill to write a play about Falstaff in love. So tonally and plot-wise, Windsor has almost nothing to do with the history plays, and the Falstaff that shows up there is more comically portrayed.
The rest of the history plays are really not terribly connected, and "historical" plays that are more considered tragedies (such as Macbeth and Hamlet and Julius Caesar) are simply not in any way associated with the tetralogies at all, and in fact can be blatantly ahistorical -- observe the clocks in Julius Caesar or the various James VI / James I flattery found in Macbeth, which is largely the reason that play was written in the first place. Like Marvel and China, Shakespeare knew how to flatter his patrons.