You probably need to develop your understanding of promises and how async JavaScript works a little.

If you have to use jQuery, you could promisify the jQuery $.ajax method (which appears to use a very old version of the promise API?), and then use that with modern JS constructs like async/await.

const pGet = (url, dataType = 'jsonp') => 
  new Promise((success, error) => $.ajax({url, dataType, success, error}))
   
const IP_URL = 'https://ipinfo.io/json'
const PREFERENCE_URL_PREFIX = 'https://www.example.com/getDomain.php?ip='

const go = async () => {
  try {
    const { ip } = await pGet(IP_URL)
    const preference = await pGet(`${PREFERENCE_URL_PREFIX}${ip}`)
    //do something with the preference
  } catch(err) {
    // handle errors
  }
}

go()

<script src="https://unpkg.com/@babel/standalone@7/babel.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>

If you don't have to use jQuery, I'd avoid it and use the fetch API provided by modern host environments.

const IP_URL = 'https://ipinfo.io/json'
const PREFERENCE_URL_PREFIX = 'https://www.example.com/getDomain.php?ip='

const go = async () => {
  try {
    const { ip } = await (await fetch(IP_URL)).json()
    const preference = await fetch(`${PREFERENCE_URL_PREFIX}${ip}`)
    //do something with the preference
  } catch(err) {
    // handle errors
  }
}

go()

<script src="https://unpkg.com/@babel/standalone@7/babel.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>

You need to return a Promise from check_saved_url. Inside of the Promise, you then need to use resolve to replace return. You can also use reject(new Error("error")) if there was an error.

async function check_saved_url() {
  return new Promise((resolve, reject) => {
    $.get("https://ipinfo.io/json", function (response) {
      current_ip = response.ip;
      location_here = response.country;

      var ajax_get_url_prefix = base_url + 'getDomain.php?ip=';
      var key_ip = current_ip + '@@@' + domain_permanent;
      var url = ajax_get_url_prefix + key_ip;

      $.get(url, function(data, status) {
        console.log("Data: " + data + "\nStatus: " + status);
        if (data.includes('0 results')) {
          // Instead of return, use resolve()
          //return 'unknown'; 
          resolve('unknown');
          // You can also pass an error with `reject(new Error("error"))`
        } else {
          // Instead of return, use resolve()
          //return data; 
          resolve(data);
        }
      });
    }, "jsonp");
  });
}

const func1 = async() => {
  return await check_saved_url();
}

const func2 = async() => {
  let domain_preference = '';
  domain_preference = await func1();
  console.log("domain_preference: ",domain_preference);
}
func2();

You can read more about promises on MDN

There is no promise returned by check_saved_url nor is $.get returning a promise. You need to wrap it all in a promise

async function check_saved_url() {
    return new Promise((resolve, reject) => {
        $.get("https://ipinfo.io/json", function (response) {
            current_ip = response.ip;
            location_here = response.country;

            var ajax_get_url_prefix = base_url + 'getDomain.php?ip=';
            var key_ip = current_ip + '@@@' + domain_permanent;
            var url = ajax_get_url_prefix + key_ip;

            $.get(url, function(data, status) {
                console.log("Data: " + data + "\nStatus: " + status);
                if (data.includes('0 results')) {
                    resolve('unknown');
                } else {
                    resolve(data);
                }
            })
                .fail(reject);
        }, "jsonp")
            .fail(reject);
    })
}

Clean the code

To have a cleaner and reusable code you can wrap $.get in a Promise

async function async_get(url) {
    return new Promise((resolve, reject) => {
        const jqxhr = $.get(url);
        jqxhr.done(() => resolve({ json: jqxhr.responseJSON, status: jqxhr.status }));
        jqxhr.fail(() => reject(jqxhr));
    })
}

async function check_saved_url() {
    const { json: response } = await async_get("https://ipinfo.io/json");
    current_ip = response.ip;
    location_here = response.country;
    var ajax_get_url_prefix = base_url + 'getDomain.php?ip=';
    var key_ip = current_ip + '@@@' + domain_permanent;
    var url = ajax_get_url_prefix + key_ip;

    const { json: data, status } = await $.get(url);
    console.log("Data: " + data + "\nStatus: " + status);
    return data.includes('0 results') ? 'unknown' : data;
}