I enabled custom-logo
for my theme and have it printed with <?php the_custom_logo(); ?>
into the header. Is there any chance to simply add some more classes to this image directly? Per default it only comes with 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