There actually was a Shakespeare. His biography is known in the roughest sketch. He married Anne Hathaway when she was pregnant. He had three children including a son named Hamnet who died young. He moved to London and made a name for himself in the theater and at the age of 50 returned to Stratford where he died two years later.
The major controversy surrounding Shakespeare is whether he wrote those plays in the first place. There is a conspiracy theory that claims that Shakespeare was merely a front for a noble who didn't want to be associated with the works and the theater. The theory probably got a lot more traction after the 1950s blacklisting where several Hollywood screenwriters (most notably Dalton Trumbo) kept working and used other people to pose for them. In other words, if Dalton Trumbo could write Roman Holiday and other screenplays and watch other screenwriters accept his awards, then some noble (Bacon, the earl of Oxford, etc.) could do the same thing in the 16th century and use Shakespeare as his front.
However, there's an underlying class snobbery in this theory. Shakespeare is respected now but in his time he was a playwright who was trying to work and make plays that he could act in. He gained popularity but most people thought that the theater was low class which is exactly what a community college Catholic from a small town could succeed in at the time.
Later on, both Shakespeare and theater became high class so the class snobs want to claim him.