When a word longer than any found so far is identified, you need to create an entirely new array (discarding any elements it had before in it), and put the new record-setting word in it. You also need to push a name to the array if the name being iterated over isn't record-setting, but is equal to the current record-setter.

const myArr = ["first", "second", "third", "fourth", "fifth", "sixth", "seven", "eighth"]

function longerStrings(arr) {
  let largeStrings = []
  let longerWord = ''
  for (let name of arr) {
    if (name.length > longerWord.length) {
      longerWord = name
      largeStrings = [name];
    } else if (name.length === longerWord.length) {
      largeStrings.push(name)
    }
  }
  return largeStrings
}
console.log(longerStrings(myArr));

Easier if you use filter instead:

const myArr = ["first", "second", "third", "fourth", "fifth", "sixth", "seven", "eighth"]
const longest = myArr.reduce((a, b) =>  a.length > b.length ? a : b);
const res = myArr.filter(el => el.length >= longest.length)
console.log(res)

That returns:

["second","fourth","eighth"]

Or, as Sash mentions below, it can be done in a single pass, although the solution is not as readable:

let max = 0;
const reducer = myArray.reduce((previous, current) => {
  if (current.length > max) {
    previous = [];
  }
  if (current.length >= max) {
    previous.push(current);
    max = current.length;
  }
  return previous;
}, []);

console.log(reducer);

You could just sort the array by length descending and then select only values which have the same length as the first element:

const myArr = ["first", "second", "third", "fourth", "fifth", "sixth", "seven", "eighth"]

const result = myArr
  .sort((a, b) => b.length - a.length)
  .filter((v, _, arr) => v.length == arr[0].length)
  
console.log(result)