Octopus refers to two of Calvin's alter egos. They each appeared in a weekday strip.
In its first appearance, the octopus was purported to "[terrorize] [a] sleepy waterfront community" by snatching tourists on a beach. This would actually be impossible for any octopus, which would be incapacitated outside of the water. An anology for Calvin really just grabbing his mother.
Here, the octopus was a cartoonlike caricature with an anthropomorphic face and character.
In its second appearance, the octopus was in its natural element performing regular octopus behavior. Threatened by an unidentified enemy, it released a cloud of ink and escaped, as an analogy to Calvin spraying Susie with pen ink at school.
The octopus was depicted realistically here, with convincing textures and acurate morphology.