You can try with this simple way by using Array.find() along with Array.some() methods.

Working Demo :

let animals = [{
    "housePets": [
    { "name": "Cat", "food" : "Fish", "description": "Long tail, sharp claws" },
    { "name": "Dog", "food" : "Biscuits", "description" : "Humans best friend" }]
}, {
  "wildAnimals": [
    { "name": "Giraffe", "food" : "Leafes", "description" : "A very long neck" },
    { "name": "Lion", "food" : "Meat", "description" : "Featured in Lion King" }]
}, {
  "insects": [
    { "name": "Ladybug", "food" : "Leafes", "description" : "Red and black" },
    { "name": "Spider", "food" : "Flies", "description" : "From the friendly neighbourhood" }]
}];

let animalsArr2 = ['housePets'];

const res = animals.find((obj) => animalsArr2.some((elem) => Object.keys(obj).includes(elem)));

console.log(res);

You can use the .find() array method to return the list that you want. For example if you want to return the wild animals list:

let animals =  [
    {
        "housePets" : [
            { "name": "Cat", "food" : "Fish", "description": "Long tail, sharp claws" }, 
            { "name": "Dog", "food" : "Biscuits", "description" : "Humans best friend" }
        ]
    },
    {
        "wildAnimals" : [
            { "name": "Giraffe", "food" : "Leafes", "description" : "A very long neck" },
            { "name": "Lion", "food" : "Meat", "description" : "Featured in Lion King" }
        ]
      
    },
    {
        "insects" : [
            { "name": "Ladybug", "food" : "Leafes", "description" : "Red and black" },
            { "name": "Spider", "food" : "Flies", "description" : "From the friendly neighbourhood" }
        ]
    }
];

const wildAnimals = animals.find((item) => item.wildAnimals);

The code above basically checks which item in your array has a property called wildAnimals and returns that item. You can read more about this method Here.

Hope that helped!

I'm gonna suppose you are forced to use this format of storing data, but I feel like I have to let you know that this isn't the best way to store info like this (you will see that it is hard to work with it). Instead of having an array of objects, you could have an object of arrays like this:

let animals = {
  "housePets": [{ "name": "Cat", "food": "Fish", "description": "Long tail, sharp claws" },  { "name": "Dog", "food": "Biscuits", "description": "Humans best friend" }],
  "wildAnimals": [{ "name": "Giraffe", "food": "Leafes", "description": "A very long neck" }, { "name": "Lion", "food": "Meat", "description": "Featured in Lion King" }]
}

With this approach, getting all house animals would be as easy as animals.housePets.
Back to your issue tho. If you want to get all house animals from this, you will have to filter the array. To filter arrays, you have to give it a condition that the element has to pass to be stored. Here you can see that only the object with the house animals has a housePets property, so we can take advantage of that:

let house_animals = animals.filter(obj => obj.housePets)[0].housePets; // [{ "name": "Cat", "food": "Fish", "description": "Long tail, sharp claws" },  { "name": "Dog", "food": "Biscuits", "description": "Humans best friend" }]

What it does is that it takes all the objects that have the "housePets" property, takes the first one (I suppose there will always be only one) and reads the property, giving you the array of animals.
To find animals that are only in the housePets section and not in others, you can do this:

let house_animals = animals.filter(obj => obj.housePets)[0].housePets;
let wild_animals = animals.filter(obj => obj.wildAnimals)[0].wildAnimals;
let insects = animals.filter(obj => obj.insects)[0].insects;
let only_house = house_animals.filter(animal => !wild_animals.includes(animal) && !insects.includes(animal); // This checks if the animal is in the wild_animals or insects array. If it isn't, it keeps it.

You can also check this question out
Hope this helped :)