Filter pagination markup. And some wp_link_pages_link.

filter-pagination-markup.php

<?php
add_filter( 'navigation_markup_template', function( $template, $class ) {
	// Set custom modifier classes.
	$class = 'pagination' === $class ? 'posts' : 'comments';

	// Modified template with custom classes.
	$template = '
	<nav class="pagination pagination--' . $class . '" role="navigation">
		<h2 class="screen-reader-text">%2$s</h2>
		<div class="pagination__items">%3$s</div>
	</nav>';

	return $template;
}, 10, 2 );

/**
 * Wraps page "links" that aren't actually links (just text) with `<span class="page-numbers">` so that they
 * can also be styled.  This makes `wp_link_pages()` consistent with the output of `paginate_links()`.
 *
 * @param  string $link Link in wp_link_pages().
 * @return string
 */
add_filter( 'wp_link_pages_link', function( $link ) {
	return 0 !== strpos( $link, '<a' ) ? "<span class='pagination__item'>{$link}</span>" : $link;
}, 5 );