Expose a custom field of a custom post type to the REST API

Modify your code as below:

/**********************************************
* Add featured video to REST API
***********************************************/
function add_featured_video_id() {
    register_rest_field(
    array('post', 'project'), 'featured_video_id',
    array(
        'get_callback'    => 'get_featured_video_id',
        'update_callback' => null,
        'schema'          => null,
        )
    );
}

Visit here, for more details on method register_rest_field,