Add the `all_membership_names` field for the `[pmpro_member]` shortcode to display a comma separated list of all the active membership level names for the current user.

pmpro-member-shortcode-field-all-memberships-names.php

<?php
/**
 * This recipe adds the custom field all_memberships_names for the [pmpro_member] shortcode
 * that will display a comma separated list of all the user's membership level(s).
 *
 * Example Usage of [pmpro_member field="all_memberships_names"]
 *
 * You can add this recipe to your site by creating a custom plugin
 * or using the Code Snippets plugin available for free in the WordPress repository.
 * Read this companion article for step-by-step directions on either method.
 * https://www.paidmembershipspro.com/create-a-plugin-for-pmpro-customizations/
 */

function my_pmpro_member_shortcode_custom_field_all_memberships_names( $r, $user_id, $field ) {

	// Check if the field is the one we want to modify.
	if ( 'all_memberships_names' !== $field ) {
		return $r;
	}

	// Get all the user's membership levels and display the level name and expiration date for each level they have.
	$levels = pmpro_getMembershipLevelsForUser( $user_id );
	$level_names = wp_list_pluck( $levels, 'name' );
	if ( ! empty( $levels ) && ! empty( $level_names ) ) {
		// Return the level names as a comma separated string.
		$r = implode( ', ', $level_names );
	} else {
		$r = '';
	}

	return $r;
}
add_filter( 'pmpro_member_shortcode_field', 'my_pmpro_member_shortcode_custom_field_all_memberships_names', 10, 3 );