If I receive an email notifying of a new issue, can I reply to the email to assign the issue to a user without logging into Jira?
With OTB JIRA no you cannot. Closest you get is a post function that automatically assigns assignee.
For a more robust solution consider JETI which allows you to use regex to search for specific terms in the header and body of the mail and you can then create a field rule to set assignee based on these rules.
You may also want to look at other Email Addons that extend JIRA to see what they offer.
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