I built a similar feature for my personal website, that consists of a new database table, WordPress admin page and some .htaccess rewrites. Unfortunately the setup is too involved with my particular installation to be useful as a plugin, but there are plenty of URL shortener plugins available already.
A quick search yields this WP plugin but I’ve never tried it. You might need to try several plugins out, but I’m sure there is one available that can provide approximately the functionality you’re looking for.