Here is my problem with this practice - most gun retailers do not do it - that is, for most gun retailers that I have been to in person, only showing them the PAL is needed. They may record the PAL information in their system, but they don't make a copy of it, and certainly they don't ask for my driver's licnce.
One retailer in the Toronto area has this practice of photocopying the PAL and driver's licence - that retailer I refuse to give them my business, which is my choice, just like it is theirs to have that practice.
Now, if one orders online, the retailer obviously will have your address and credit card number on file, and the billing address entered needs to match what it says on the credit card in order for the transaction to go through.
In person, however, I fail to understand why have this practice. If there is an issue of identity theft etc., you would think the bulk of the firearms vendor would adopt the same policy. But the vast majority of them don't.