How can I use dynamic paths that all load the same file?

Install the URL Rewrite module then go into the IIS Management application. Click the website in question. Open URL Rewrite. Create a new rule.

Requested URL: Does not match the pattern
Using: Regular expressions
Pattern: \/(?!.*index|$)
Ignore case: true
Action: Rewrite
Rewrite URL: index.php?tk=1
Append query string: true
Stop processing of subsequent rules: true

This assumes you use index.php as your script file. If you use default.php, just change the index to default in the regex.  If you had this as your index.php:

<?php

echo array_key_exists("REQUEST_URI", $_SERVER) ? $_SERVER["REQUEST_URI"] : "No URI";

?>

then you went to https://www.mydomain/com/whatever, you would get this output:

/whatever

Which you can then use to query the database for the appropriate URL to push.  In my rewrite URL example I did index.php?tk=1 so you can look for $_REQUEST["tk"] to know this is an attempt to do a saved URL, not simply normal use of the site.  Obv change that to whatever "fingerprint" format you wish.