Create new folder and upload files to custom folder via wp_handle_upload

This isn’t a working example but should help,

Use wp_upload_dir() to define the default uploads folder and append a username to it using wp_get_current_user.

$wp_upload_dir =  wp_upload_dir();
$user_folder = wp_get_current_user();

// The actual folder
$custom_upload_folder= $wp_upload_dir['basedir'] . $user_folder->display_name;
//make the dir
mkdir($custom_upload_folder);

You would need to employ security and code that checks for errors/user capabilities and sets the correct file/folder permissions. Also have a look at https://codex.wordpress.org/Filesystem_API which employs mkdir.