Technically you could disable the a built-in notification for when a ticket is Updated, and you could create an Automation rule to send a custom email which includes Epic Name and any other fields you wanted:
Some caveats (making me think you don't really want to do this):
But it's possible. And being able to send to an arbitrary email address can be useful if you need to send notifications to a Team email alias that isn't linked to a Jira user.
On the other hand, maintenance for this doesn't scale well.
Oh a few other thoughts:
Unfortunately, currently, it is not possible to edit mail templates for notifications.
You can check the feature request JRACLOUD-7266 to check more information. I will paste below the information that Atlassian provided in the feature request:
In the spirit of transparency, we are going to close this feature request for JIRA Cloud. It's been 18 months since we last provided an update and we do not have any plans to introduce support for custom email templates in JIRA Core or JIRA Software. (This is how we've been interpreting this feature request, as editing velocity templates has never been possible for JIRA Cloud).
Please check the information above and let me know if it helps.
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