The script has about the same requirements as WordPress, so you can easily install it alongside your WordPress installation in a subdirectory (eg https://example.com/shortener/) or in a subdomain (eg https://shortener.example.com/).
You could then put a link from your WordPress site’s navigation menu to the script site (and vice versa) and try your best to mimic the designs as much as possible.
You wouldn’t need to do any PHP coding at all, but the two will run totally independently of each other.
But if you want to “integrate” the two somewhat: eg load the script inside of WordPress, let the script use WordPress’ user registration system and database, etc. — then you’re going to have to do some custom PHP coding to make that happen.
That “custom PHP coding” could be a simple WordPress plugin (or even custom theme template) loading the full external script on a WordPress page… or it could involve a complete reworking of the script itself to make it become a fully integrated part of your WordPress site and database.
