Setting Cookies

Solution with cookie [wordpress]

You will need jQuery cookie plugin.
Create javascript file “themechanger.js”:

(function($){

    var theme = $.cookie("styles");
    var themehref = document.getElementById(theme_styles.rel_id);

    setTheme = function(mode){
        if (theme == mode)
            return;
        if (mode == 'light') {
            $.cookie('styles', mode);
            themehref.href = theme_styles.light;
        }
        else if (mode == 'dark') {
            $.cookie('styles', mode);
            themehref.href = theme_styles.dark;
        }
    }
})(jQuery);

In functions.php enqueue styles and javascript:

add_action('init', 'load_styles', 15);
function load_styles() {

    $current_style="/styleSheetLightTheme.css";
    if ( isset($_COOKIE['styles']) && $_COOKIE['styles'] == "dark")
        $current_style="/styleSheetDarkTheme.css";
    wp_register_style( 'stylename', get_stylesheet_directory_uri() . $current_style );
    wp_enqueue_style( 'stylename');

    wp_enqueue_script( 'jquery_cookie', get_template_directory_uri() . '/js.cookie.js', ['jquery'] );
    wp_register_script( 'theme_script', get_template_directory_uri() . '/themechanger.js', ['jquery', 'jquery_cookie'] );
    //
    // object "theme_styles" will store links to style, used in javascript
    wp_localize_script( 'theme_script', 'theme_styles', [
        'light' => get_stylesheet_directory_uri() . '/styleSheetLightTheme.css',
        'dark' => get_stylesheet_directory_uri() . '/styleSheetDarkTheme.css',
        'rel_id' => 'stylename-css',
    ]);
    wp_enqueue_script('theme_script');
}

To change theme style use javascript function setTheme()

<a href="https://wordpress.stackexchange.com/questions/311547/javascript:;" onClick="setTheme('dark'); return false;">Dark Theme</a><br>
<a href="https://wordpress.stackexchange.com/questions/311547/javascript:;" onClick="setTheme('light'); return false;">Light Theme</a>

Local storage [for your code]:

<html>
<head>
  <link id="my_style" rel="stylesheet" type="text/css" href="https://wordpress.stackexchange.com/questions/311547/styleSheetLightTheme.css">
</head>
<body>
    <script>
        var theme = localStorage.getItem('styles');
        var themehref = document.getElementById('my_style');
        var base_url="styleSheet{{mode}}.css";

        // default value
        if(!theme)
            localStorage.setItem('styles','light');

        if (theme == 'dark')
            themehref.href = base_url.replace('{{mode}}', 'DarkTheme');
        else
            themehref.href = base_url.replace('{{mode}}', "LightTheme");

        function setTheme(mode){
            if ( (mode != 'light' && mode != 'dark'))
                return;

            theme = localStorage.getItem('styles');
            // don't change if it's already active
            if (mode == theme)
                return;
            if (mode == 'light') {
                localStorage.setItem('styles', 'light');
                themehref.href = base_url.replace('{{mode}}', 'LightTheme');
            }
            else if (mode == 'dark') {
                localStorage.setItem('styles', 'dark');
                themehref.href = base_url.replace('{{mode}}', 'DarkTheme');
            }
        }
    </script>

    <a href="https://wordpress.stackexchange.com/questions/311547/javascript:;" onClick="setTheme('dark'); return false;">Dark Theme</a>
    |
    <a href="https://wordpress.stackexchange.com/questions/311547/javascript:;" onClick="setTheme('light'); return false;">Light Theme</a>
</body>
</html>