Query custom post type and showing its content

You are not closing the second while loop.
You’ll probably want something like this:

<?php 
 /*
 * Template Name: Testimonials
 */
 ?>

<?php get_header(); ?>

<div class="container">
    <div class="content-page">
        <h1><?php wp_title(); ?></h1>
        <title><?php wp_title(); ?></title>
        <?php
         $query = new WP_Query( array('post_type' => 'testimonials', 'posts_per_page' => 5 ) );
         while ( $query->have_posts() ) : $query->the_post(); ?>

        <?php endwhile; ?>
    </div>
</div>

<?php 
$args = array( 'post_type' => 'testimonials', 'posts_per_page' => 10 );
$the_query = new WP_Query( $args ); 
?>
<?php if ( $the_query->have_posts() ) : ?>
    <?php while ( $the_query->have_posts() ) : $the_query->the_post(); ?>
        <h2><?php the_title(); ?></h2>
        <div class="entry-content">
            <?php the_content(); ?> 
        </div>
    <?php endwhile; ?>
<?php else:  ?>
    <p><?php _e( 'Sorry, no posts matched your criteria.' ); ?></p>
<?php endif; ?>
<?php wp_reset_postdata(); ?>

<?php get_footer(); ?>

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