No. Making a copy of the recipe does not change the Event Name of the Trigger. Since the Event Name remains unchanged, the Webhooks URL does not change.
Webhooks URL follows the following format:
https://www.workato.com/webhooks/rest/<User Id>/<Event Name>
Why is this important?
If you have two recipes running at the same time with the same Webhooks URL, both recipes will trigger off when the same event happens.