You can use Promise.all() like:

const artists = ['first', 'second'];

const promises = artists.map(artist => fetch(url+artist));

Promise.all(promises)
.then(response => {
  // handle response
})
.catch(err);

or

const returnedData = await Promise.all(promises).catch(err);

Looking at the pieces one at a time, setArtists must indeed by async and should probably run the validation currently (since the validations are not interdependent).

There's no need for the artists global in the containing scope. In fact, having it will encourage errors.

// clarifying... input is a comma-delimited string describing artists
// validate each one with spotify, and return an array of the valid artists
async function setArtists(input) {
  let parsedInput = input.value.split(",")
  let promises = parsedInput.map(validateArtist);
  let results = await Promise.all(promises);  // validate all of them at once
  // return just the valid inputs, not the nulls
  return results.filter(r => r);
}

The validateArtist method mixes async styles. Here it is with uniform style and with a clearer objective: to just validate and return an artist...

// given the params of an artist, lookup on spotify and return a
// a promise that resolves to the input artist if valid, null otherwise
async function validateArtist(s) {
  const token = localStorage.getItem("Token")
  const url = "https://api.spotify.com/v1/search?type=artist&q=" + s
  console.log(url)
  const response = await fetch(url, {
      "method": "GET",
      "headers": {
        'Accept': 'application/json',
        'Content-Type': 'application/json',
        "Authorization": "Bearer " + token,
      }
  });
  if (response.status === 401) {
    refreshToken(s)
  }
  const searchedResults = await response.json();
  const isArtist = searchedResults.artists.items.length != 0;
  // notice, no side-effects here, resolve to the input artist or null
  return isArtist ? s : null;
}

Lastly, your caller must be async also, and await the setArtists result...

async function theCaller() {
  // notice - no need for a global. now it's a local here...
  let artists = await setArtists(document.getElementById("artistiPreferiti"))
  var user = {
    username: document.getElementById("username").value,
    email: document.getElementById("email").value,
    password: document.getElementById("password").value,
    gustiMusicali: document.getElementById("gustiMusicali").value,
    artistiPreferiti: artists
  } 
  // ...

As a side note, there might be valuable stuff in the artist array that spotify returns, some super set of the data you queried with. If you'd rather keep the results around, you can make up a policy to pick the first matching artist returned in the array of spotify matches...

// updating the functional comment:
// given the params of an artist, lookup on spotify and return a
// a promise that resolves to the first found artist or null if no matches are found

  .then(searchedResults => {
    let items = searchedResults.artists.items;
    return items.length ? items[0] : null;
  });