Use an alternative to `bp_core_get_user_displayname()` to get the user's display name without HTML tags when sending a friendship request.

bp-custom.php

<?php
/**
 * Custom function file.
 *
 * Put a file called `bp-custom.php` into your /wp-content/plugins folder.
 */

// Exit if accessed directly.
if ( ! defined( 'ABSPATH' ) ) {
	exit;
}

/**
 * Uses the WP User display name to avoid filters in `bp_core_get_user_displayname()`
 *
 * @since 1.0.0
 *
 * @param array $args Email tokens.
 * @return array      Email tokens.
 */
function nohtmltag_in_friends_email_subject( $args = array() ) {
	if ( isset( $args['tokens']['friend-requests.url'], $args['tokens']['initiator.name'], $args['tokens']['initiator.id'] ) && $args['tokens']['initiator.id'] ) {
		$user = get_user_by( 'id', $args['tokens']['initiator.id'] );

		if ( isset( $user->ID ) ) {
			/*
			 * Edit the title of the Email from the WP Admin > Emails screen replacing
			 * [{{{site.name}}}] New friendship request from {{initiator.name}} with:
			 * [{{{site.name}}}] New friendship request from {{initiator.first_and_last_name}}
			 *
			 * That being said, I sugget to always use `$user->display_name` instead of 
			 * `$user->first_name . ' ' . $user->last_name` as some users are not setting these fields
			 * and BuddyPress guesses these from the display name using what's before the space as the first
			 * name & what's after the space as the last name.
			 */
			$args['tokens']['initiator.first_and_last_name'] = $user->first_name . ' ' . $user->last_name;
		}
	}

	return $args;
}
add_filter( 'bp_before_send_email_parse_args', 'nohtmltag_in_friends_email_subject', 10, 1 );