Q:

Why was Shakespeare's work in the mirror universe no different than in the normal universe?

In the mirror universe, Archer and his team captured the USS Defiant of the normal universe. After analysing the data of its computers, they found that the works of Shakespeare in both universes were same.
Human natures in both universes were completely different. One would expect to be able to assume the same for Shakespeare, too. How can two different people having two different personalities write the same work of literature?

A:

Not everyone in the mirror universe is the opposite of his or her counterpart.
In "Mirror, Mirror", the TOS episode that introduced the mirror universe, both versions of the Halkans were pretty much the same, intensely idealistic pacifists.  We saw Hitler in the opening credits of the Enterprise mirror episode; presumably he was just as evil but more successful. And Kirk observed that Spock was "a man of integrity in both universes".
And it's plausible that the two universes share a common ancestry, with some event in history that went one way in one universe and the other way in the other.  Shakespeare may have lived before the two universes diverged.
On the other hand, in "Dark Mirror", by Diane Duane, is TNG mirror universe novel, Picard reads the mirror universe version of Shakespeare's "Merchant of Venice". According to TV Tropes:

In the Mirror Universe Shylock gets his pound of flesh because no one
  would really think you can really get a pound of flesh without
  shedding blood. They weigh it, it is too much, and they laughingly say
  he can take some of it back.

But that's not canonical, and it contradicts the scene in Enterprise (written later) where we find that Shakespeare is the same in both universes.

