Octodad is an octopus, pretending to be a human. He's a mimic octopus.
Doesn't look at all like a mimic octopus. Still, the comparison works.
In Dadliest Catch, during the flashback level you have to move items off a map. The map shows that the ship is sailing from Great Britain towards Iceland, hinting that "the war" might refer to the Cod Wars, an actual conflict between the UK and Iceland over fishing rights in the North Atlantic.
The clumsy controls, and especially the Multiplayer makes a bit of sense when you learn that octopi have distributed intelligence, meaning that each arm has enough neurons to have a mind of it's own that is equal parts controlled directly and "autopiloted" based on information gained from the octopus' sensory organs.
It's pretty laughable that no one notices you're an octopus in a suit. But then again, your "wife" appears normal at the shark tank until you get caught in Fujimoto's trap.
Fridge Horror
It's certainly fun for players to stagger around struggling with the poor controls, but if one looks at what Octodad is actually doing as he tries to finish human tasks—stumbling around, repeatedly smacking his kids and his wife, "mumbling" incoherently, destroying furniture, etc.—and looks while under the idea that he's a normal human husband, he rather resembles a drunken lout. Furthermore the fact that his wife is so okay with that could be chalked up to being a Stepford Smiler in a more serious work. Luckily, this is not a serious work.
Fridge Logic
The game tracks your "health" using a suspicion bar, which shows you how much others suspect you are an octopus. This makes sense in the first few levels, where you are trying to do normal tasks for your family, but later on, in Octodad's secret lab, the suspicion meter fills up when you're hit by lasers, and the only other guy in the room with you already knows you're an octopus.
In Dadliest Catch, Octodad falls over like he died when the Suspicion Meter fills up fully. Though this does make a bit of sense when you get hit by Fujimoto or his cleavers.
Fridge Brilliance: He's an octopus. If his cover is blown, what's the first instinct of an octopus? Hide.
So, uh...where did the kids come from? (This question is lampshaded in Dadliest Catch, but not answered.)
How about artificial insemination? Why not? It would also explain Stacy.
How in the world does a sea creature with gill survive for years on land? Yes, an octopus can survive for a little while out of water (in captivity, they have a tendency to break out of their containments and wander around at night), but that's only as long as they keep their gills moist. So does Octodad just go dunk his head in the sink whenever he starts feeling dried out?
Maybe his suit is specially made to help him keep moist on land without drawing suspicion as to why hes constantly dripping.
One of his wife's lines in the aquarium mentions that she doesn't know what he keeps in the basement, but respects his privacy. That something could be a pool he takes a dip in every so often.
In the "Dadliest Catch" level, Home Sweet Home, how do Scarlet, Tommy or Stacy not notice the GIANT, LOUD machine Chef Fujimoto was using to try and suck in their husband/father. If you look to the left during this scene there are white dots coming from there to Octodad, meaning Scarlet, Tommy or Stacy can see him during this, so are none of them the least bit concerned of him seconds away from getting sucked into a propeller, or at least the chef loudly referring to him as "octopus"?
, an actual conflict between the UK and Iceland over fishing rights in the North Atlantic.