I have set up a specific email address as a target to receive email notifications when an update is made to a ticket. When the recipient replies to the email notification, rather than having their reply threaded to the original ticket, the email reply creates a new ticket. I would expect their reply to be added as an entry to the original ticket that the notification was sent from.
Why do replies to email-target notifications create a new ticket? How can I ensure customer email replies are added to the first ticket instead of creating a new ticket?
The target notification system is not designed to thread to an existing ticket and it is expected behavior that a new ticket would be created.
To allow this end user's email replies to thread to the original ticket (instead of creating a new ticket), you will want to stop using targets as your notification method. Instead, you should CC the end-user to your ticket so that they receive the CC notification. If they reply to that notification, their reply is added to the ticket as a public comment.
See Configuring CC permissions and notifications for more information on how you can customize your CC notification settings.