When the Payment Txn status changes from Matching Start to Matching Complete, potential contacts to link with are located using the Matching Rules on the Merchant Facility.
For the Nonprofit Success Pack (NPSP), a final check is made to ensure the First Name is ALWAYS a match as well. If the First Name does not match, then a new contact is created for the same household.
The reason for the First Name having significance in these Matching Rules is due to a "Household" individual contacts often sharing things like emails, address and phone/mobile numbers.