Show user's email address in place of the login name in the "Your are logged in as" message on the PMPro checkout page.

gettext_checkout_email_in_logged_in_message.php

<?php
/**
 * Replace user's login displayed to email address in the "You are logged in as"
 * on the PMPro checkout page.
 *
 * 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_gettext_checkout_email_in_logged_in_message( $translated_text, $original_text, $domain ) {
	global $current_user;
	if ( 'paid-memberships-pro' === $domain ) {
		if ( 'You are logged in as <strong>%s</strong>. If you would like to use a different account for this membership, <a href="%s">log out now</a>.' === $original_text ) {
			$translated_text = 'You are logged in as <strong>' . esc_html( $current_user->user_email ) . '</strong><span style="visibility: hidden; display: none;">%s</span>. If you would like to use a different account for this membership, <a href="%s">log out now</a>.';
		}
	}
	return $translated_text;
}
add_filter( 'gettext', 'my_pmpro_gettext_checkout_email_in_logged_in_message', 10, 3 );