Remove the post format prefix from the name property of the term objects created by wp_get_object_terms().

function _post_format_wp_get_object_terms( $terms ) {
	foreach ( (array) $terms as $order => $term ) {
		if ( isset( $term->taxonomy ) && 'post_format' === $term->taxonomy ) {
			$terms[ $order ]->name = get_post_format_string( str_replace( 'post-format-', '', $term->slug ) );
		}
	}
	return $terms;
}