You’ll have the best results resolving customers when they have an incentive to provide your systems with their correct customer numbers. The best incentive is better service through recognizing and remembering that customer.
Consider Amazon.com. The usefulness of its recommendation engine and comprehensive online order history is a powerful motivator to use the same login for every purchase. Be careful of marketing promotions or financial rewards tied to new-customer or previous-customer status. They can lead customers to split their accounts (where the customer uses a slightly different name or address so as to claim a new-customer discount multiple times). Or it could lead less-than-scrupulous call center reps to merge customer records (where, in order to close more sales, the rep offers all callers the “loyal returning customer” benefit by using the same “gold” buyer record for each call, repeatedly mangling the address.)