Yes, it's possible - but you'd have to do it in code.
The contact form that the email links to contains a "tracer" id which tells your system which email it is a response to. If the contact form is on a web site, then just add it as a query string:
www.MyDomain.com/contactForm.aspx?tid=c7e00cfd-1a3d-41ac-83d4-8066f4622a01
This may help
Passing Query String Parameters in WordPress URL - Razorlight Media[
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Your code looks up the tracerID in your DB, and sends an email with a new ID value.
I'd suggest using GUID's for this as it's trivial to generate them and they shouldn't repeat.