The modern sense is very old, from the 13th century, but in Shakespeare's time it was supplemented by several other senses that later died out.
Two are particularly important. When Antipholus of Syracuse is described three times in The Comedy of Errors as being in a rage, the word means "madness", not "anger": Adriana says of him that "till this afternoon his passion Ne'er brake into extremity of rage" (V.i.48).
This is the sense required when Cordelia and the Doctor discuss Lear's rage: "The great rage ... is killed in him" (King Lear, V.vii.78) or Juliet worries about waking in the tomb in a "rage" (Romeo and Juliet, IV.iii.53).
There is also a very positive sense to be noted. When Talbot talks about "great rage of heart" (Henry VI Part 1, IV.vii.11) or Hotspur says he was "dry with rage and extreme toil" (Henry IV Part 1, I.iii.30) they are not talking about anger but about warlike spirit, martial ardour.