"Shakespeare" has never been much more than a name that was attached to a very diverse group of plays--the information about him is scarce, and, over the years, since we needed him to be much more than we had, speculation filled in all the gaps--Plays, for that matter, are not novels or poetry--they are entertainment, and often, they are an agglomeration of material from a variety of sources--dialog, sketches, and routines, then, as now, were often   created by the performers themselves and interpolated into the plays--

Manuscripts come from all manner of places, and have to be reworked considerably (often without the blessing, or even the knowledge, of the original writer) in order to work on the stage--so who wrote it? Compare it with the question of authorship of hollywood films--in many cases, several teams of writers have reworked the material, and the writing credit is given as per contract, rather than based on who wrote the words finally used--why would it have been different then than now? I doubt that it was--