Sample Plugin - Create new user role and new post type and allow to create the custom post type with those users who have that specific user role.

sample-plugin.php

<?php
<?php
/**
 * Plugin name: Sample Plugin
 */

/**
 * User Role Specific Post Type
 *
 * @package User Role Specific Post Type
 */

if( ! class_exists( 'Sample_Plugin' ) ) :

	/**
	 * Sample Plugin
	 */
	class Sample_Plugin {

		/**
		 * Member Variable
		 *
		 * @var instance
		 */
		private static $instance;

		/**
		 * Initiator
		 *
		 * @since x.x.x
		 */
		public static function get_instance() {
			if ( ! isset( self::$instance ) ) {
				self::$instance = new self;
			}
			return self::$instance;
		}

		/**
		 * Constructor
		 * 
		 * @since x.x.x
		 */
		public function __construct() {
			add_action( 'admin_init', array( $this, 'add_user_role' ), 999 );
			add_action( 'init', array( $this, 'register_post_type' ) );
		}

		/**
		 * Add user role
		 *
		 * @return void
		 */
		function add_user_role() {
			$role = get_role( 'doc_writer' );

			if ( ! $role ) {
				add_role( 'doc_writer', 'Doc Writer' );
				$role = get_role( 'doc_writer' );
			}

			$role->add_cap( 'level_0' );
			$role->add_cap( 'read' );

			// Primitive capabilities used within map_meta_cap().
			$role->add_cap( 'read' );
			$role->add_cap( 'delete_doc_writer' );
			$role->add_cap( 'read_private_doc_writers' );
			$role->add_cap( 'delete_private_doc_writer' );
			$role->add_cap( 'delete_published_doc_writer' );
			$role->add_cap( 'delete_others_doc_writer' );
			$role->add_cap( 'edit_private_doc_writer' );
			$role->add_cap( 'edit_published_doc_writer' );

			// Meta capabilities.
			$role->add_cap( 'edit_doc_writer' );
			$role->add_cap( 'edit_doc_writers' );
			$role->add_cap( 'edit_other_doc_writer' );
			$role->add_cap( 'read_doc_writer' );

			// Primitive capabilities used outside of map_meta_cap().
			$role->add_cap( 'edit_others_doc_writer' );
			$role->add_cap( 'publish_doc_writer' );
			$role->add_cap( 'read_private_doc_writer' );

			$role->add_cap( 'upload_files' );
			$role->add_cap( 'delete_posts' );

			// Enable to delete attachments.
			$role->add_cap( 'delete_others_posts' );
		}

		function register_post_type() {

			$labels = array(
				'name'               => __( 'Docs', 'text-domain' ),
				'singular_name'      => __( 'Doc', 'text-domain' ),
				'add_new'            => _x( 'Add New Doc', 'text-domain', 'text-domain' ),
				'add_new_item'       => __( 'Add New Doc', 'text-domain' ),
				'edit_item'          => __( 'Edit Doc', 'text-domain' ),
				'new_item'           => __( 'New Doc', 'text-domain' ),
				'view_item'          => __( 'View Doc', 'text-domain' ),
				'search_items'       => __( 'Search Docs', 'text-domain' ),
				'not_found'          => __( 'No Docs found', 'text-domain' ),
				'not_found_in_trash' => __( 'No Docs found in Trash', 'text-domain' ),
				'parent_item_colon'  => __( 'Parent Doc:', 'text-domain' ),
				'menu_name'          => __( 'Docs', 'text-domain' ),
			);

			$args = array(
				'labels'              => $labels,
				'hierarchical'        => false,
				'description'         => 'description',
				'taxonomies'          => array(),
				'public'              => true,
				'show_ui'             => true,
				'show_in_menu'        => true,
				'show_in_admin_bar'   => true,
				'menu_position'       => null,
				'menu_icon'           => null,
				'show_in_nav_menus'   => true,
				'publicly_queryable'  => true,
				'exclude_from_search' => false,
				'has_archive'         => true,
				'query_var'           => true,
				'can_export'          => true,
				'rewrite'             => true,
				'capabilities' => array(
		            'edit_post' => 'edit_doc_writer',
		            'edit_posts' => 'edit_doc_writers',
		            'edit_others_posts' => 'edit_other_doc_writer',
		            'publish_posts' => 'publish_doc_writer',
		            'read_post' => 'read_doc_writer',
		            'read_private_posts' => 'read_private_doc_writers',
		            'delete_post' => 'delete_doc_writer',
	           ),
				'supports'            => array(
					'title',
					'editor',
				),
			);

			register_post_type( 'doc', $args );
		}

	}

	/**
	 * Kicking this off by calling 'get_instance()' method
	 */
	Sample_Plugin::get_instance();

endif;