See https://wordpress.org/support/topic/specific-post-type-not-working-with-wpapitrue/#post-8857621 for more information.

plugin.php

<?php
/**
 * Plugin Name: Add a list of Jetpack related posts to the bottom of each post in the Rest API Posts Embeds shortcode.
 * Plugin URI: http://jeremy.hu/
 * Description: Add a list of Jetpack related posts to the bottom of each post in the Rest API Posts Embeds shortcode.
 * Author: Jeremy Herve
 * Version: 1.0.0
 * Author URI: https://jeremy.hu
 * License: GPL2+
 */
/**
 * Add a list of Jetpack related posts to the bottom of each post in the Rest API Posts Embeds shortcode.
 *
 * @see https://wordpress.org/support/topic/specific-post-type-not-working-with-wpapitrue/
 *
 * @param string $article     Article layout.
 * @param object $single_post Array of information about the post.
 */
function jeherve_add_color_post( $article, $single_post ) {

	// Build our site URL.
	$parsed_url = parse_url( $single_post->guid->rendered );
	if ( isset( $parsed_url['scheme'], $parsed_url['host'] ) ) {
		$site_url = $parsed_url['scheme'] . '://' . $parsed_url['host'];
	} else {
		return $article;
	}

	$post_rest_url = sprintf(
		'%1$s/wp-json/wp/v2/posts/%2$d',
		esc_url( $site_url ),
		absint( $single_post->id )
	);

	// Hash that post URL to cache it in a transient.
	$post_rest_hash = substr( md5( $post_rest_url ), 0, 21 );

	// Look for data in our transient.
	$data_from_cache = get_transient( 'jeherve_post_embed_' . $post_rest_hash );
	if ( false === $data_from_cache ) {
		$post_data = wp_remote_get( esc_url_raw( $post_rest_url ) );

		if (
			is_wp_error( $post_data )
			|| 200 != $post_data['response']['code']
			|| empty( $post_data['body'] )
		) {
			return $article;
		}

		$post_data_body = wp_remote_retrieve_body( $post_data );
		$post_info = json_decode( $post_data_body );

		// Cache the result.
		set_transient( 'jeherve_post_embed_' . $post_rest_hash, $post_info, 10 * MINUTE_IN_SECONDS );
	} else {
		$post_info = $data_from_cache;
	}

	// Do we have an array of Related Posts?
	if ( is_array( $post_info->{'jetpack-related-posts'} ) && ! empty( $post_info->{'jetpack-related-posts'} ) ) {
		$related_posts = '<ul>';
		// Loop through each related post to build an unordered list.
		foreach ( $post_info->{'jetpack-related-posts'} as $related_post ) {
			$related_posts .= sprintf(
				'<li><a href="%1$s">%2$s</a></li>',
				esc_url( $related_post->url ),
				esc_html( $related_post->title )
			);
		}
		$related_posts .= '</ul>';

		// Append our list of related posts to each article.
		return $article . $related_posts;
	}

	// Final fallback.
	return $article;
}
add_filter( 'jeherve_post_embed_article_layout', 'jeherve_add_color_post', 20, 2 );