The supernatural figures in The Comedy of Errors are purely an excuse to ignore the complexity of reality. There is no single occurrence that cannot be explained by some perfectly natural (if bizarre) reasoning, but characters are quick to point to the fates, dreaming, madness, and general supernatural stuff (devils, sorcery, witchcraft) in order to explain the strangeness of their situations. The supernatural stands in as a convenient explanation for what seems inexplicable, given the implausible truth that under-girds the entire play.
The supernatural functions only as a plot device in the play. No character that attributes anything to the supernatural ever gives a viable reason for doing so. This is only a way to explain the Ephesian strangeness – and move the plot along.
Each character that invokes the supernatural does it as cover up for some personal weakness. Egeon blames the supernatural fates for his miserable condition; S. Antipholus’s inability to deal with and understand his reality leave him quick to jump to some explanation outside of himself; and Adriana would rather believe her husband is possessed than deal with the possibility that he just might not love her anymore.