Get taxonomy name for the current post

You should no longer use the legacy function parameter format. Instead, as the documentation says, use the get_terms( $args ) format:

Since 4.5.0, taxonomies should be passed via the ‘taxonomy’ argument
in the $args array:

$terms = get_terms( array(
  'taxonomy'   => 'post_tag',
  'hide_empty' => false,
) );

And as for getting only the terms assigned to the current or a specific post, you can either use the object_ids parameter with get_terms(), or simply use get_the_terms().

So for examples:

$post_id = get_the_ID();

$countries = get_terms( array(
    'taxonomy'   => 'country',
    'object_ids' => $post_id, // set the object_ids
) );

// Or just use get_the_terms():
$countries = get_the_terms( $post_id, 'country' );