The staff at the paint store got rather confused and couldn't really figure out what to do with me, or how to take my money, when I refused to join their club, saying that I'd rather pay the 20% extra. We only managed to proceed when my wife filled in the form.

Turns out you could just lie and fill out anything on the form.

One strategy I heard about is to get a local group of like-minded individuals together, throw all of your loyalty cards in a communal pile then take one back randomly. That mixes the signal up for any kind of targeted tracking while also getting you the discounts. Or you can just say "I forgot my card" and use the store copy that is usually at the register (at least for groceries).

If the company uses phone numbers to lookup accounts, (Local-Area-Code)-8675-309 almost always works.

https://en.wikipedia.org/wiki/867-5309/Jenny

Don't worry, depending on the state, they have plenty of other ways to personally identify you. No need.