Custom Logo URL | Help me print the URL of the custom logo I inserted into my theme

Use wp_get_attachment_image_src to get the image properties and URL:

$logo = get_theme_mod( 'custom_logo' );
$image = wp_get_attachment_image_src( $logo , 'full' );
$image_url = $image[0];
$image_width = $image[1];
$image_height = $image[2];

Edit: Adding more information based on your comment.

The URL it points to is simply made with home_url:

esc_url( home_url( "https://wordpress.stackexchange.com/" ) );

Another thing often forgotten when using this method is honoring the content of the alt if it’s set in the media:

$alt = get_post_meta( $logo, '_wp_attachment_image_alt', true );
if ( empty( $alt ) ) {
    // Use site title if no alt is provided.
    $alt = get_bloginfo( 'name', 'display' );
}

I’d encourage looking at the source of get_custom_logo as it would use most of this.

It’s also worth mentioning if you’re developing this theme for release on WordPress.org/themes – it’s a requirement to use the_custom_logo or get_custom_logo over rolling out your own solution.

For this reason there is a filter for the HTML output in get_custom_logo, which might be best to use and replace the bits you need. The function get_custom_logo (used by the_custom_logo) also handle a bit more like multisite and ensuring the placeholder is there for the customizer preview. The usage would be something like this:

add_action( 'get_custom_logo', function( $html ) {
    $new_url="href="" . get_theme_mod( 'custom_logo_url', esc_url( home_url( "https://wordpress.stackexchange.com/" ) ) ) .'"';
    $new_link_class="class="box-link"";
    $html = str_replace( 'href="' . esc_url( home_url( "https://wordpress.stackexchange.com/" ) ) . '"', $new_url, $html );
    $html = str_replace( 'class="custom-logo-link"', $new_link_class, $html );
    return $html;
} );

That would read a value of the theme_mod custom_logo_url which could come from a customizer control, and changes the class from .custom-logo-link to .box-link in your example.

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