I don’t think Aizawa owns cats but he is seen with them a lot. For those who are mystified over Aizawa, his full name is Shota Aizawa and he is a Japanese manga character. In English terms that makes him a cartoon character.
His Fandom CV contains no mention of Aizawa owning domestic cats. However, there are images online of him with cats.
For example, on Pinterest, there are several drawings of him. He is very recognisable by his wild, copious, light-coloured scarf. It seems to be his trademark item of clothing.
In one drawing he is leaning against a wall while a cat saunters by. He is looking at the cat. In another, written entirely in Japanese, he is with four cats and an associate. I don’t know what they are saying but clearly they like having cats around.
In another he is on a poster labelled N.275 – Encounter, Part 2, headed ‘My Hero Academia’. There is a tabby cat in the foreground. The cat is well drawn. And in another he is holding a kitten at his shoulder and smiling.
All these drawings might indicate that he owns cats but it is unclear to me. Although clearly there is a definite connection though between Aizawa and cats.
Reddit.com has a fandom page and one page is headed with the question ‘When is it stated how much Aizawa likes cats? Personally, I have never seen it stated that he likes cats but the drawings strongly suggest that he does.
But I may be barking up the wrong tree and have misinterpreted the drawings and the questioning.
It appears to me that the artists who created the manga cartoons wanted to make the connection between this cartoon character and domestic cats to create a certain image. Perhaps an image of kindness to go with his skills as a fighter. To create a full and rounded character. I am guessing and speculating as I must.
P.S. Fandom says about him: (相あい澤ざわ消しょう太た Aizawa Shōta?), also known as the Erasure Hero: Eraser Head (イレイザー・ヘッド Ireizā Heddo?), is a Pro Hero and the homeroom teacher of U.A. High School’s Class 1-A.
The manga world is alient to me and many Europeans. It is like trying to understand another language. It is mystifying as I said.