This add-in gives you full Polylang support in WordPress customizer. By full support I mean that you customize each language site differently. Original from https://github.com/soderlind/customizer-polylang

customizer-polylang.js

/**
 *
 */
/* global wp, jQuery */
/* exported PluginCustomizer */
var PSPolyLang = (function (api, $) {
	'use strict';

	var component = {
		data: {
			url: null,
			languages: null,
			current_language: null,
		}
	};

	/**
	 * Initialize functionality.
	 *
	 * @param {object} args Args.
	 * @param {string} args.url  Preview URL.
	 * @returns {void}
	 */
	component.init = function init(pll) {
		_.extend(component.data, pll);
		if (!pll || !pll.url || !pll.languages || !pll.current_language) {
			throw new Error('Missing args');
		}

		api.bind('ready', function () {
			api.previewer.previewUrl.set(pll.url);

			var languages = pll.languages;
			var current_language = pll.current_language;
			var current_language_name = '';

			var html = '<span style="position:relative;left:38px">Language: </span>';
			html += '<select id="pll-language-select" style="position:relative; left: 35px; top: 1px; padding: 0 2em 0 .5em;">';
			for (var i = 0; i < languages.length; i++) {
				var language = languages[i];
				var selected = (language.slug === current_language) ? 'selected' : '';
				current_language_name = (language.slug === current_language) ? language.name : '';
				html += '<option ' + selected + ' value="' + language.slug + '">' + language.name + '</option>';
			}
			html += '</select>';
			$(html).prependTo('#customize-header-actions');


			$('body').on('change', '#pll-language-select', function () {
				var language = $(this).val();
				var old_url = window.location.href;
				window.location.href = updateQueryStringParameter(window.location.href, 'lang', language);
			});
		});

		function updateQueryStringParameter(uri, key, value) {
			var re = new RegExp("([?&])" + key + "=.*?(&|$)", "i");
			var separator = uri.indexOf('?') !== -1 ? "&" : "?";
			if (uri.match(re)) {
				return uri.replace(re, '$1' + key + "=" + value + '$2');
			} else {
				return uri + separator + key + "=" + value;
			}
		}
	};

	return component;
}(wp.customize, jQuery));

CustomizerPolylang.php

<?php

/**
 * This add-in gives you full Polylang support in WordPress customizer. By full support I mean that you customize each language site differently.
 * Minor adaptation from https://github.com/soderlind/customizer-polylang to fit into the Hello Roots logic.
 */

namespace SayHello\Theme\Plugin;

class CustomizerPolylang
{
	public function run()
	{

		if (!function_exists('pll_current_language') || !function_exists('pll_default_language')) {
			return;
		}

		/**
		 * Force "The language is set from content" (in Language->Settings->URL modifications)
		 */
		$options = get_option('polylang');
		if (isset($options['force_lang']) && 0 !== $options['force_lang']) {
			$options['force_lang'] = 0;
			update_option('polylang', $options);
		}

		add_filter('pll_preferred_language', '__return_false');

		add_action('customize_controls_enqueue_scripts', [$this, 'addLangToCustomizerPreviewer'], 9);
		add_action('wp_before_admin_bar_render', [$this, 'onWpBeforeAdminBarRender'], 100);
		add_action('admin_menu', [$this, 'onAdminMenu'], 100);

		$theme_stylesheet_slug = get_option('stylesheet');
		$option_types = ['blogname', 'blogdescription', 'site_icon'];

		add_filter('option_theme_mods_' . $theme_stylesheet_slug, [$this, 'onOptionThemeModsGet'], 10, 1);
		add_filter('pre_update_option_theme_mods_' . $theme_stylesheet_slug, [$this, 'onOptionThemeModsUpdate'], 10, 2);

		foreach ($option_types as $option_type) {
			add_filter('pre_option_' . $option_type, [$this, 'onWpOptionGet'], 10, 3); // get_option hook.
			add_filter('pre_update_option_' . $option_type, [$this, 'onWpOptionUpdate'], 10, 3); // update_option hook.
		}

		$this->registerPolylangColumnStrings('options_footer_columns', ['title', 'text']);
		$this->registerPolylangColumnStrings('options_footer_logos', ['image', 'url']);
	}

	/**
	 * Helper to fetch custom customizer db content.
	 *
	 * @return mixed Customizer array or false.
	 */
	protected function getCustomCustomizerOption()
	{
		$current_language = pll_current_language();
		$theme_slug       = get_option('template');
		$option_prefix    = \str_replace('-', '_', $theme_slug);
		$option_name      = $option_prefix . '_customizer_polylang_settings_' . $current_language;

		return get_option($option_name, false);
	}

	/**
	 * Helper to update custom customizer db content.
	 *
	 * @param mixed $data Data to insert.
	 *
	 * @return bool Success.
	 */
	protected function updateCustomCustomizerOption($data)
	{
		$current_language = pll_current_language();
		$theme_slug       = get_option('template');
		$option_prefix    = \str_replace('-', '_', $theme_slug);
		$option_name      = $option_prefix . '_customizer_polylang_settings_' . $current_language;

		return update_option($option_name, $data);
	}

	/**
	 * Helper
	 *
	 * @return bool If the current language is the default language.
	 */
	protected function currentLangNotDefault()
	{
		return pll_current_language() !== pll_default_language();
	}

	/**
	 * Check the custom db field on get_option hook to be able to return custom language value.
	 * If the current language is default, then return from default wp option
	 *
	 * @param bool   $pre_option This is false. If something else is returned wp exits the check in db and uses this value.
	 * @param string $option Option name asked for.
	 * @param mixed  $default Default value, second args when asking for options.
	 *
	 * @return mixed
	 */
	public function onWpOptionGet($pre_option, $option, $default)
	{

		// If not the default language, then skip the custom check and wp will the use default options.
		if ($this->currentLangNotDefault()) {
			$data = $this->getCustomCustomizerOption();

			// Found the custom option. Move on.
			if (is_array($data) && isset($data['options']) && isset($data['options'][$option])) {
				return $data['options'][$option];
			}
		}

		return $default;
	}

	/**
	 * Update the custom db field on get_option hook.
	 * If the current language is not default, then return old value to prevent from saving to default wp option.
	 *
	 * @param mixed  $value The new, unserialized option value.
	 * @param mixed  $old_value The old option value.
	 * @param string $option Option name.
	 *
	 * @return mixed
	 */
	public function onWpOptionUpdate($value, $old_value, $option)
	{
		// Fetch custom option db field.
		$data       = $this->getCustomCustomizerOption();
		$theme_slug = get_option('template');
		// If false, the field hasn't been created yet, so it must be created.
		if (false === $data) {
			$data = [
				'template' => $theme_slug,
				'mods'     => [],
				'options'  => [],
			];
		}

		// Make sure the options array exists. We are going to use it soon.
		if (!isset($data['options'])) {
			$data['options'] = [];
		}

		$data['options'][$option] = $value;

		// Update option value in custom db field. (Not necessary to save for default language since it uses default wp option fields for values when get option).
		$this->updateCustomCustomizerOption($data);

		// If the current language is not the default language, prevent saving to option table by passing the old value back. It will then exit after the filter.
		if ($this->currentLangNotDefault()) {
			return $old_value;
		}

		return $value;
	}

	/**
	 * Check the custom db field on get_option customizer field option name hook to be able to return custom language value.
	 * Parse arguments with default wp customizer values to make sure all are present in the return.
	 *
	 * @param array $value The customizer settings.
	 *
	 * @return array
	 */
	public function onOptionThemeModsGet($value)
	{
		$data = $this->getCustomCustomizerOption();

		if (isset($data['mods']) && is_array($data['mods']) && !empty($data['mods'])) {
			$value = wp_parse_args($data['mods'], $value);
		}

		// Remove members with integer key from mods.
		foreach ($value as $key => $mod) {
			if (is_numeric($key)) {
				unset($value[$key]);
			}
		}

		return $value;
	}

	/**
	 * Update custom customizer option.
	 * If the current language is not default, then return old value to prevent from saving to customizer wp option.
	 *
	 * @param mixed $value The new, unserialized option value.
	 * @param mixed $old_value The old option value.
	 */
	public function onOptionThemeModsUpdate($value, $old_value)
	{

		$current_data = $this->getCustomCustomizerOption();
		$theme_slug   = get_option('template');

		$data = [
			'template' => $theme_slug,
			'mods'     => isset($current_data['mods']) ? $current_data['mods'] : [],
			'options'  => isset($current_data['options']) ? $current_data['options'] : [],
		];

		if (is_array($value) && !empty($value)) {
			foreach ($value as $key => $val) {
				$data['mods'][$key] = $val;
			}
		}
		$this->updateCustomCustomizerOption($data);

		if ($this->currentLangNotDefault()) {
			return $old_value;
		}

		return $value;
	}

	/**
	 * If Polylang activated, set the preview url and add select language control
	 */
	public function addLangToCustomizerPreviewer()
	{

		$handle    = 'dss-add-lang-to-template';
		$src       = get_stylesheet_directory_uri() . '/assets/scripts/customizer-polylang.js';
		$deps      = ['customize-controls'];
		$version   = rand();
		$in_footer = 1;
		wp_enqueue_script($handle, $src, $deps, $version, $in_footer);
		$language = (empty($_REQUEST['lang'])) ? pll_current_language() : $_REQUEST['lang'];

		if (empty($language)) {
			$language = pll_default_language();
		}

		if (!empty($_REQUEST['url'])) {
			$current_url = add_query_arg('lang', $language, $_REQUEST['url']);
		} else {
			$current_url = add_query_arg('lang', $language, pll_home_url($language));
		}
		wp_add_inline_script(
			$handle,
			sprintf(
				'PSPolyLang.init( %s );',
				wp_json_encode(
					[
						'url'              => $current_url,
						'languages'        => get_transient('pll_languages_list'),
						'current_language' => $language,
					]
				)
			),
			'after'
		);
	}

	/**
	 * Append lang="contrycode" to the customizer url in the adminbar
	 *
	 * @return void
	 */
	public function onWpBeforeAdminBarRender()
	{
		global $wp_admin_bar;
		$customize_node = $wp_admin_bar->get_node('customize');
		if (!empty($customize_node)) {
			$customize_node->href = add_query_arg('lang', pll_current_language(), $customize_node->href);
			$wp_admin_bar->add_node($customize_node);
		}
	}

	/**
	 * Append lang="contrycode" to the customizer url in the Admin->Apperance->Customize menu
	 *
	 * @return void
	 */
	public function onAdminMenu()
	{
		global $submenu;
		$parent = 'themes.php';
		if (!isset($submenu[$parent])) {
			return;
		}
		foreach ($submenu[$parent] as $k => $d) {
			if ('customize' === $d['1']) {
				$submenu[$parent][$k]['2'] = add_query_arg('lang', pll_current_language(), $submenu[$parent][$k]['2']);
				break;
			}
		}
	}

	public function registerPolylangColumnStrings($option_name, $fields)
	{
		$columns    = get_option($option_name);
		$theme_name = wp_get_theme()->get('Name');
		if (!empty($columns)) :
			for ($i = 0; $i < $columns; $i++) :
				foreach ($fields as $field) {
					pll_register_string($option_name . '_' . $i . '_' . $field, get_option($option_name . '_' . $i . '_' . $field), $theme_name, true);
				}
			endfor;
		endif;
	}
}