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In The Handmaiden (2016), Kouzuki is a sadistic pervert who controls Hideko's and her aunt's lives. His most powerful secret weapon is the basement, later revealed to be a torture chamber. Hideko is so afraid of it that she makes conman Fujiwara promise to give

her a vial of poison guaranteed to kill her quickly in the event she is caught by her uncle and taken back to the basement.

We later get to see the torture chamber, after Fujiwara is caught by Kouzuki and made to tell him details about Hideko.
In that scene, we also see glimpses of a giant octopus moving in a large water tank.
You can see it in this scene (not for sensitive viewers):

Why did Kouzuki keep a giant octopus in his torture chamber?

The aunt tried to run away and paid for it with her life, Hideko would rather commit suicide than be brought back into the basement - why?

The octopus is the answer. It tells us, what happened down there, without having to show it on screen.

From Philip Gowman: Park Chan-wook talks about Handmaiden, octopuses and more, London Korean Links, 2016.10.24:

Does Park have an obsession with the octopus?

No. The boneless sea creature that Choi Min-sik ate alive in Oldboy is 낙지. The writhing object in the basement in Handmaiden is 문어. Two different words in Korean. [...] In Handmaiden the Korean audience will associate the octopus in the basement with a famous Japanese erotic print. That means they know what happens in the basement without Park having to film what is depicted in the print.

Miranda Chance wrote in Pleasure / pain and The Handmaiden -- does a film have to be difficult to be good? for thecontentjournal:

But not only is the octopus the director’s personal nod to Oldboy (2003), a cult classic, but it conveniently features heavily in Japanese pornography, the most famous of which, Hokusai’s ‘The Dream of the Fisherman’s Wife’, makes an appearance.

According to its wikipedia article,

it depicts a young ama diver entwined sexually with a pair of octopuses.

The picture itself is also featured in the movie, according to the Cannes 2016 Review.

From TV tropes

the octopus in the mansion's basement, just like the significantly smaller edible octopus in Oldboy arguably symbolizes imprisonment.

But it seems more of a nod to Oldboy where actor Choi Min-sik in a famous scene gobbled down alive cephalopod on camera.