I think it would. With real cards we shuffle the deck and then pick some cards from the top of the top of the deck. This is what you'll probably be doing with the shuffle function, thereby modelling the real world use.

With Math.Random(), you're randomly picking a card from an un-shuffled deck. The key here is randomness (which is not really random btw). So, although this isn't modelled after the real world use, the end result is the same.

I would suggest Math.Random() because it will, although not significantly, be faster than using _.shuffle's (Fisher–Yates) algorithm.

One simple approach is to just shuffle the array of indices and then use that to get both your arrays in the corresponding order:

const mp3 = ['sing.mp3','song.mp3','tune.mp3','jam.mp3'];
const ogg = ['sing.ogg','song.ogg','tune.ogg','jam.ogg'];

const indices = _.shuffle([0, 1, 2, 3]);

const shuffledMp3 = indices.map(idx => mp3[idx]);
const shuffledOgg = indices.map(idx => ogg[idx]);

console.log(shuffledMp3, shuffledOgg);

<script src="https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.21/lodash.min.js"></script>

However, you shouldn't need to do this. Your question only arises because you're storing the data in an unhelpful way. Since the elements of the two arrays are clearly linked to each other, you shouldn't store them as 2 arrays, but as a single array whose elements are objects holding both filenames as properties. This enables you to get what you want with a single shuffle:

const files = [{mp3: 'sing.mp3', ogg: 'sing.ogg'}, {mp3: 'song.mp3', ogg: 'song.ogg'}, {mp3: 'tune.mp3', ogg: 'tune.ogg'}, {mp3: 'jam.mp3', ogg: 'jam.ogg'}];

const shuffled = _.shuffle(files);

console.log(shuffled.map(file => file.mp3));
console.log(shuffled.map(file => file.ogg));

<script src="https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.21/lodash.min.js"></script>

(It's also worth noting that if all your file names are as predictably formed as in your example, even this is unnecessary - you can just store the "raw" filenames: ['sing', 'song', 'tune', 'jam'] and add whatever file extension you need at the point of needing it. But perhaps your real data is not so consistent.)

The de-facto unbiased shuffle algorithm is the Fisher–Yates (aka Knuth) Shuffle.

You can see a great visualization here.

function shuffle(array) {
  let currentIndex = array.length;

  // While there remain elements to shuffle...
  while (currentIndex != 0) {

    // Pick a remaining element...
    let randomIndex = Math.floor(Math.random() * currentIndex);
    currentIndex--;

    // And swap it with the current element.
    [array[currentIndex], array[randomIndex]] = [
      array[randomIndex], array[currentIndex]];
  }
}

// Used like so
let arr = [2, 11, 37, 42];
shuffle(arr);
console.log(arr);

Try sorting like this snippet:

console.log( [
    { some: 1 },
    { some: 2 },
    { some: 3 },
    { some: 4 },
    { some: 5 },
    { some: 6 },
    { some: 7 },
  ]
  .sort( () => Math.random() - 0.5) );

In reponse to Martin Omanders comment: here's a shuffle method according to the Fisher-Yates algorithm

const result = document.querySelector("pre");
for (let i=0; i<20; i+=1) {
  result.textContent += 
    JSON.stringify(shuffleFisherYates([0,1,2,3,4,5,6,7,8,9])) + '\n';
}

function shuffleFisherYates(array) {
  let i = array.length;
  while (i--) {
    const ri = Math.floor(Math.random() * i);
    [array[i], array[ri]] = [array[ri], array[i]];
  }
  return array;
}

<pre></pre>

Which may be condensed to a one liner (note: this one liner will not compile in the Google Closure Compiler with level advanced):

const shuffle = array => 
  [...Array(array.length)]
    .map((el, i) => Math.floor(Math.random() * i))
    .reduce( (a, rv, i) => ([a[i], a[rv]] = [a[rv], a[i]]) && a, array);
const result = document.querySelector("pre");
for (let i=0; i<100; i+=1)
  result.textContent +=
   JSON.stringify(shuffle([0,1,2,3,4,5,6,7,8,9])) + '\n';

<pre></pre>