Here's a functional way, returning a new set:

const set = new Set(['a', 'b', 'c'])
const arr = ['d', 'e', 'f']
const extendedSet = new Set([ ...set, ...arr ])
// Set { 'a', 'b', 'c', 'd', 'e', 'f' }

If you start out with:

let array = [
    {name: "malcom", dogType: "four-legged"},
    {name: "peabody", dogType: "three-legged"},
    {name: "pablo", dogType: "two-legged"}
];

And you want a set of, say, names, you would do:

let namesSet = new Set(array.map(item => item.name));

It's not using a Set, but may solve your problem.

If you want to make sure that an array can only exist in a collection once, and you want instant lookups you can use a hash by abusing its keys. They do not necessarily need a value. You can set them to undefined, to true, or something else if it makes you sleep better at night 😛.

Inspired by: javascript search array of arrays

It is halfway tempting to make a wrapper, but it is also simple enough to handle a small job.

const hash = {};

hash[[1, 2, 3]] = undefined;
hash[[3, 4]] = undefined;
hash[[3, 4]] = undefined;

console.log({hash});
console.log("hash has [3, 4]?", hash.hasOwnProperty([3, 4]));

For me, this will work fine, because I am checking if a coordinate exists in a collection. Unfortunately from this stackoverflow it seems that Set cannot be the answer to this problem.

Coincidentally it seems to implicitly solve your problem of not calling toString on the array which this seems to do under the hood. Personally I don't care for hash.hasOwnProperty.

A More Elegant Option

You could even write a wrapper to your code like:

class ArraySet extends Set {
  add(arr) {
    super.add(arr.toString());
  }
  has(arr) {
    return super.has(arr.toString());
  }
}

const arraySet = new ArraySet();

arraySet.add([1, 2]);
arraySet.add([3, 4]);
arraySet.add([3, 4]);

console.log("ArraySet has [3, 4]?", arraySet.has([3, 4]));