Short answer

What you exactly want is impossible. Objects cannot have multiple keys.

My suggestion

Have an object where the value of each key is an array. If an animal have multiple sound, you'll only have to push it into the array.

var
animals = ['Cow', 'Cow', 'Dog', 'Cat'],
sounds = ['Moo', 'Oink', 'Woof', 'Miao'],
result = {};

for(i = 0; i < animals.length; i++) {
    result[animals[i]]
        ? result[animals[i]].push(sounds[i])
        : result[animals[i]] = [sounds[i]];
}

Fiddle

JS Fiddle of all 3 options: http://jsfiddle.net/ysLvfxmd/2/

You want a hash/associative array, loop through and create one by index, however you cannot have duplicate keys in a hash so you may want to explore other data structures.

Pure Hash

var animals = ['Cow', 'Cow', 'Dog', 'Cat'];
var sounds = ['Moo', 'Oink', 'Woof', 'Miao'];
var hash = {};

for(i = 0; i < animals.length; i++) {
    hash[animals[i]] = sounds[i];
}

console.log(hash);

This will only show Cow once, if you want to have pairs of this with non-unique keys you need to make an array of hashes

Array Of Hashes

var animals = ['Cow', 'Cow', 'Dog', 'Cat'];
var sounds = ['Moo', 'Oink', 'Woof', 'Miao'];
var arr = [];

for(i = 0; i < animals.length; i++) {
    var hash = {};
    hash[animals[i]] = sounds[i];
    arr.push(hash);
}

console.log(arr);

However, note that the indexes of the array are now numeric but you can search through it to find your values.

Third Option:

Hash Of Arrays (Best)

//Paul Rob's suggestion hash of arrays

var animals = ['Cow', 'Cow', 'Dog', 'Cat'];
var sounds = ['Moo', 'Oink', 'Woof', 'Miao'];
var hashOfArr = [];

for(i = 0; i < animals.length; i++) {
    if(!hashOfArr[animals[i]]){
        hashOfArr[animals[i]] = [];
    }
    hashOfArr[animals[i]].push(sounds[i]);
}
console.log(hashOfArr);