Wordpress - How to add CSS class to custom logo?

WordPress provide a filter hook to custom logo customization. The hook get_custom_logo is the filter. To change logo class, this code may help you.

add_filter( 'get_custom_logo', 'change_logo_class' );


function change_logo_class( $html ) {

    $html = str_replace( 'custom-logo', 'your-custom-class', $html );
    $html = str_replace( 'custom-logo-link', 'your-custom-class', $html );

    return $html;
}

Reference: How to change wordpress custom logo and logo link class

Here's one suggestion how we might try to add classes through the wp_get_attachment_image_attributes filter (untested):

add_filter( 'wp_get_attachment_image_attributes', function( $attr )
{
    if( isset( $attr['class'] )  && 'custom-logo' === $attr['class'] )
        $attr['class'] = 'custom-logo foo-bar foo bar';

    return $attr;
} );

where you adjust the classes to your needs.

As you found yourself the_custom_logo relies on get_custom_logo, which itself calls wp_get_attachment_image to add the custom-logo class. The latter function has a filter, wp_get_attachment_image_attributes which you can use to manipulate the image attributes.

So what you could do is build a filter that checks if the custom-logo class is there and if yes add more classes.