Check if current user has a post and that post has any term/s from a specific custom taxonomy outside the loop

This can be done using WP_Query using author and tax_query.

Something like this:

$args = array(
    'post_type' => 'post',
    'author' => get_current_user_id(),
    'tax_query' => array(
        array(
            'taxonomy' => 'custom-taxonomy',
            'operator' => 'EXISTS'
        ),
    ),
);
$query = new WP_Query( $args );

And then check if posts are returned through this query.

Please note that this code is not tried or tested and may contain syntax errors.