Got the answer after chatting with MS customer support: The offer gives single user license of Office 2013 and it doesn't overwrite previous license (its not an upgrade). One can continue to use old Office 2010 licenses.
Here's the chat transcript:
You are now chatting with 'Steven M'.
Steven M: Thank you for contacting Microsoft Store Chat! How can I
help you today?
Sachin Shekhar: I have 3 user Office 2010 license which I activated after
October 19, 2012. If I upgrade it to Office 2013 with the offer, would I
get 3 user Office 2013 license? Or, each 3 users need to go thru the
process to get 3 single user licenses?
Steven M: You will get a new Office 2013 licence for 1 Machine
Steven M: Please see the link bellow for all the details and terms
Steven M: Office 2013 terms
Sachin Shekhar: Would all 3 users be eligible for offer?
Sachin Shekhar: I have seen the terms, but there's no mention of
multi-user license there..
Steven M: So do you have 3 different product keys for Office 2010?
Sachin Shekhar: No. I have one product key which is valid for 3 users.
Steven M: Then you will benefit from 1 Office 2013
Sachin Shekhar: Means, my 3 user license would be trimmed down to 1
user license. Right? or, other 2 users can continue to use Office
Steven M: No, you will have 3 Office 2010 and 1 Office 2013
Sachin Shekhar: Unable to understand. Are you saying the free upgrade
doesn't overwrite the previous license?
Steven M: No, it gives you an extra product key for Office 2013, it's
not an upgrade
Sachin Shekhar: Ok. Thanks.
Steven M: Thank you for chatting with us at Microsoft Store. We highly