Was once a normal octopus monster, until one day it hitched a ride on one of the Seafolk while they were marching to land.
It sees "Caped Baldy" on a storefront TV once on land, and likes how he looks enough to mimic him as a way to fit in. Turns out that it did well, as most people can't seem to tell the difference between the two - not because the mimicry is good, but because most people just honestly don't remember, or recognize, or care about Caped Baldy in general.
Gets into various disputes with other monsters once on land, and was strong enough to kill them.
Genos starts wondering why Saitama is getting credit for monsters he didn't kill, and it doesn't take very long before he finds out it's this strange octopus (all it took was watching a televised fight). He lets it slide, until strange rumors start to pop up...
"I swear, I think I saw Caped Baldy slithering around out in my backyard last night!",
"Caped Baldy: Is he as slimy on the outside as he is on the inside?",
"This homeless bald guy in a stupid costume was rooting through my trash this morning", etc.
Saitama and Genos work together to track down and stop the octopus, but when they all meet the two find out that the octopus means no harm. It's just friendly, curious, confused, and seems to admire Saitama. The two decide that killing the octopus is not an option.
Genos and Saitama first tried to deal with Octopunch by returning it to the sea whenever it found its way into the city. That was effective for a while, but Octopunch would always tire of the sea and come back.
Eventually Octopunch became familiar enough with Saitama, Genos, and the cities that it breaks into the apartment first thing after coming back from the ocean. At that point, Saitama and Genos just decide it'd be easier to keep it than to keep making trips to the beach.
As Octopunch spends more time around Saitama and Genos and their friends, it learns how to be stronger, and how to be a better person and hero.
Some other tibits:
Saitama was the one who first called it "Octopunch", and the name stuck
Octopunch also responds to "octopus", "Caped Baldy", and "Sensei"
Genos sometimes call it sensei by mistake when he just sees it out of the corner of his eye.
Its "baldy face" never changes expression
It loves to get into Genos' sardines and Caped Baldy merch.
Genos is happy to let Octopunch play with the more cheap and durable merchandise, but they fight over the sardines, and Genos hides the fragile and expensive merch somewhere at Kuseno's
Sonic is one of the few people who can tell the real Saitama apart from his octopus mimic, but he doesn't get it until he tries to fight it first.
He figures it out pretty quickly after Octopunch grabs his sword with only half a "leg"
It's better at video games than Saitama
Cats love it, for all the wrong reasons (Seafood, yum yum!)
The reason it can survive on land for days on end? Simple, it's just holding its breath!
Octopunch takes residence in the tub when nobody else is using it
A good "uncle" to the Genoslings. One arm for each bot!
Octopunch is very strong and durable for an octopus - it mimiced from the best, after all. Its suction cups are particuarily difficult to escape from.
Even when the octopus and real saitama are next to each other most people just assume that the octopus is a human
"Oh, I didn't know you had a kid! He looks just like you!"
King jokes about liking the octopus better than Saitama. Sonic says the same thing, but means it
Genos feels weird when he cooks seafood for dinner whenever Octopunch is around. Octopunch itself doesn't care so long as it's tasty
More than once Genos had almost cooked Octopunch by accident while it was hiding inside a pot, or the oven
Not that it would've harmed it, Octopunch is pretty sturdy
It can understand spoken speech and writing, but will feign ignorance where convenient (which is most of the time). It eventually learns how to write, but it can't spell very well.
Somehow, its handwriting is neater than Saitama's
One thing Genos and Saitama struggles with early on after adopting Octopunch is teaching it that it isn't ok to climb on strangers.
Lecturing it didn't work, so they had to keep bribing it with sardines until it stopped being a habit.
They also had to teach Octopunch that stealing was wrong, which was made more difficult by the fact that it didn't have a concept of property or stealing to begin with.
For a long time they had to be careful about Octopunch following them into stores.
Once it DOES learn that stealing is wrong, it starts to regularly offer things in exchange for various goods, first with random objects, then with coins and paper money only.
Nobody is sure where the money comes from, but there’s a lot of it (and the truth is that Octopunch is just very good at finding loose change).
Genos at this point gets a random allowance from Octopunch three times each day - one for breakfast, lunch, and dinner.
One day Saitama catches Octopunch leafing through the rent money Genos gave him. Octopunch has never seen Saitama look at him with that much intensity before.
Octopunch quickly made amends by putting back the money in its rightful place, and hasn't touched it since.
Another day, Saitama can't be bothered to go to another dull HA meeting, so Genos and Octopunch go. Genos tries to pretend everything is normal and everyone else is just like,
"Is there something different about Caped Baldy?"
"I know, he's normally never this attentive..."
(King knows, but he doesn't say anything)
Courtesy of shenanigans by @reptile-ruler, @rayadraws, @jaccident, @nyam, @chronomincer, @thornless-roses, and me
Saw this image during a lecture yesterday and I’ve had this idea stuck in my head ever since. I didn’t realize that those were numbers in circles and not eyes until I looked up the image for reference!