The chorus, as a dramatic device, originates with the Greek theatre.  As Enotes states:

The Greek chorus comments on themes, and shows how an ideal audience might react to the drama. The chorus also represents, on stage, the general population of the particular story.. . .In many of these plays, the chorus expressed to the audience what the main characters could not say, such as their hidden fears or secrets. The chorus often provided other characters with the insight they need.

For a theatrical world that presented, in its Tragedies, the struggles of  kings, queens, gods and goddesses, the chorus provided a real point of connection for the audience, which was comprised, for the most part, of ordinary Greek citizens.

Shakespeare used a character called Chorus in his plays Romeo and Juliet and Henry V, but these "characters" were not involved in the plot of either play.  The Elizabethan Chorus, as defined in the Enotes description of Henry V's Chorus is:

usually a single actor who recites the play's prologue and epilogue, apologizing for any defects the play might have and begging the audience's forbearance.  Sometimes the Chorus also fills in details that cannot be presented onstage and comments on the action of the play.

And this definition holds true for both plays in which Shakespeare used a Chorus.  In Romeo and Juliet, the Chorus speaks the Prologue and Epilogue and also introduces Act II, while in Henry V, the Chorus introduces all five Acts.

So, while it is true that Enobarbus seems a somewhat detached and objective observer of the action of the play, and does much to advise Antony, he doesn't fit the structural definition of a Chorus as used on the Elizabethan stage or by Shakespeare himself in other plays.

Shakespeare often had characters in his plays who seemed to stand outside the action and observe, or provide thematic commentary -- Jaques in As You Like It, The Duke in Measure for Measure, Kent and/or The Fool in King Lear, and, most interestingly, Prospero (who stands outside and inside the action simultaneously) in The Tempest.  Enobarbus would fit more in the vein of these characters of Shakespeare's plays.

Since he did it at least twice, create and use a character called "Chorus," we should assume that if Shakespeare had wanted a character to serve as a Chorus in Antony and Cleopatra, he would simply have put a Chorus in.

For more on the chorus and Enobarbus, please follow the links below.