This is a demonstration of how to add different favicons manually in custom wordpress theme using functions.php

functions.php

<?php
//It is not super complete but it is ok for the start
// Remember that $favicon_path will change according to your assets directory structure
function add_favicons() {
    // declare the path variables
    $favicon_path = get_template_directory_uri() . '/assets/img/favicon.ico';
    $apple_touch_icon_72x72_path = get_template_directory_uri() . '/assets/img/apple-touch-icon-72x72.png';
    $apple_touch_icon_114x114_path = get_template_directory_uri() . '/assets/img/apple-touch-icon-114x114.png';
    
    // echo to the wp_head anchor to append to it
    echo '<link rel="shortcut icon" href="' . $favicon_path . '">';
    echo "\n";
    echo '<link rel="apple-touch-icon" sizes="72x72" href="' . esc_url($apple_touch_icon_72x72_path) . '">';
    echo "\n";
    echo '<link rel="apple-touch-icon" sizes="114x114" href="' . esc_url($apple_touch_icon_114x114_path) . '">';
}
add_action('wp_head', 'add_favicons');