There is nothing supernatural about the story, no one is a cat. Author just seems to love having parallels with her characters and cats, just a fun quirk about the manhwa! She made Yeongha resemble a cat in the way he acts and the guy with white hair resembles the white cat with two black spots in his storyline (this will be more obvious in later chapters). Idk if you were only confused about the cat thing or the story in general but if you have any question I don’t mind answering it!
this is gonna be long sorry in advance
The cats are an extra in the story. Sang-ho and Young-ha went to school together. Sang-ho was popular and a jock while Young-ha was shy and didn't talk or interact with anyone (everything also indicates that Young-ha is neurodivergent, hence many of his problems relating to other people). The only one he paid attention to, the only one he had a friendship with, was Sang-ho, since he did and followed everything Sang-ho told him. When they finished school, they drifted apart, and Sang-ho's father fell ill and died. Because of this, his early twenties weren't stend like the typical college life. One day, they met again, and while talking and walking, they fed some stray cats. Sang-ho noticed that all the cats were attracted to Young-ha. He explained that it was because he works at a cat cafe (and was therefore used to cats). While talking, he told Sang-ho that he felt his boss liked him and asked Sang-ho what it was like to have sex with a man (knowing that Sang-ho was gay). He asked if Sang-ho could show him what it was like between men. Sang-ho replied that he was a virgin and didn't have much experience, explaining that because of everything that happened to his father, he hadn't had time for sex. The next day, Sang-ho is at university (where there are many stray cats on campus) and runs into Young-ha, who also attends the same university. They start talking, and Sang-ho feels comfortable around Young-ha because they are both 28 and still in university, which makes him feel less alone. When the topic of his boss comes up, he tells Young-ha that he'll accompany him to work to see if it's true that the boss likes him. He goes along and realizes that the boss doesn't have feelings for Young-ha; he's just being affectionate and already has a boyfriend. Sang-ho is impressed that all the cats at the cafe are attracted to Young-ha and wants to start working there, which he does.
They begin spending more time together, rekindling their friendship, and, intrigued, they decide to have sex together. Young-ha wants to know what it would be like to have sex with a man (he doesn't like being with other people and only feels comfortable with Sang-ho), and Sang-ho wants to lose his virginity. Obviously, there are flashbacks to their school days, and it's clear they were attracted to each other and there was more than just friendship between them, but they're too young and naive to realize it.
At the same time, Young-ha has a sunbae in is class who is gay, and he asks him what it's like to have sex with a man (he's spending more and more time with Sang-ho; they touch each other, although they haven't gone all the way yet, and he wants another opinion). The Sunbae tells him he'll explain what it's like, but with a condition. The Sunbae used to date another guy in their class, but one day they broke up, and he saw him with a girl. However, he still likes this other guy, and he's still pursuing him, so he asks Young-ha to pretend to be his boyfriend.
We still don't know if anything actually happened between the Sunbae and Young-ha, or if they're just pretending. In this episode, Sang-ho is jealous and doubtful about whether something happened between Young-ha and his Sunbae especially since he also said that the Sunbae was his friend, and Sang-ho is Young-ha's only friend, which makes him jealous.
It's simply that they're being very foolish, not realizing that they have feelings for each other, that they're attracted to each other. The cat theme is an extra, since a stray cat always appears whenever Young-ha is around. It's also somewhat of an analogy for how Young-ha is with everyone; he keeps his distance, doesn't like being surrounded by people, like a stray cat. But when Sang-ho appears, Young-ha changes and gravitates around him all the time.

I feel stupid (or I am) I read everything and yet I dont understand anything, like the character is a cat or anything like that, why the cats feel so important in the story.. someone can explain the plot to me ?