@Echonic I can help you here I think. (I love symbolism way too much)
To expand on what
@kleiner said, her father likes freedom and the ability to spread his wings. He's not a bad person, though you could argue he does eventually become indirectly neglectful, but Koto never really seems to judge him for it and later in life emulates his behavior more or less. That's why the mother thinks that, in her obsession for the orderly, that the cat got what it deserved because it lived its life as it pleases instead of adhering to some kind of order.
If you notice, the dad doesn't say that he dislikes fish or finds them repulsive. He says that he pities them more or less for being trapped, whereas the mom actively detests the cats by thinking they're filthy animals and treats them like filth. The author's doing their best to show how the Dad's compassionate and kind, and that he doesn't hate or really loathe animal/lifestyle, but finds some pitiable and sad, whereas the mom actively detests other forms of existence.
But even moreso, if you notice the fish represent the Kisaragi household, in that it's strict and orderly. You live your life trapped in a confined existence as show for others what a good family you are without any freedom of your own. They're supposed to put their life on display and be confined to their own small space with no wiggle room or freedom.
The cats, on the other hand, represent the Ichikawa household and Kosuke's and Miya's situation, in that they're too attached to Keiko after her death, which leads to their suffering. The mother would find the Ichikawas detestable, seeing the fact Kosuke can't control his daughter's behavior or his interactions with her daughter, but the father, who ran off with a younger woman, probably would hold great sympathy and empathy and try to help them if he could, which is the example Koto decides to follow in trying to "bury" the memory of Keiko so they can move on like she buried the mother cat.
The one piece I can't quite pick up is that Koto's mother looks at her breasts before she says she looks like a cat. I don't quite know specifically what to do with it. Could it be some sense of worry as she reaches maturity/womanhood as she feels she wants to control Koto all to herself and the possibility of her seeing men scares her? Or is it some kind of jealousy for the attractiveness of youth that she longs for, or maybe some sense that Koto would be a better mother than her, which causes both guilt and envy? Maybe it's the desire for some kind of sexual or romantic freedom that she feels she doesn't have, so she wants to take it away from Koto. I'm not sure yet, but I'm sure it'll be clarified later.