I have had at least three Safeway cards issued to me without having filed any paperwork. Each time the card was issued the clerk asked whether I already had a card. I responded in the negative. The clerk pulled a card out from under the till or a drawer, scanned the card to apply a discount, and then handed me the card to keep.
In fact, you can often keep the activated card (at all kinds of stores) even if they do ask you to fill in some papers. Just ask to take the forms home to fill them in and return later and take the card, too.
This is true, and your buying instruments (unless you always pay cash) or even your time of purchase/store choice will correlate out those three cards as all "you". We are very much creatures of habit and this is what Safeway and people like Target rely upon.
Now without the baby sitting co-op to randomize data I've offered my card to people ahead of me in line who don't have one, but again the gas rewards are changing that equation as well.
I've long since avoided Safeway (going on two decades), but back when I both visited the store and had a card, but under a different name (usually some variant of "Fred Flintstone" or "Mickey Mouse"), I found that my actual name eventually ended up on the receipt if I every paid by check.
Lesson: don't pay by check.
Lesson: don't patronize stores with shopper surveillance cards.
Most recent notable experience: a few visits to Border's Books, in which the store's "loyalty card" was pushed ever more vigorously. I responded "no, the NSA's already listening to my phone calls and reading my emails, no need for them to get my reading list as well" (this was years before Snowden, though after the AT&T listening room was revealed). The cashier was impressed by my integrity, and commented as much, as I paid cash for my purchase.
The last time I was asked if I had their card was during the final week of the store's liquidation sale. And I was thinking 1) why even bother and 2) where will that data go after the company shuts down (it turns out it was bought by Barnes & Nobel, which may itself go the way of Borders before too long).
Many stores let you enter your phone number on a keypad if you've forgotten your loyalty card at home. If you don't feel like signing up for a store's card to get a discount, just try entering 867-5309 (from the Jenny song) in your local area code. 99% of the time it works!
For anyone who asked my phone number i always used 555-1212 which was the 'universal' number for directory assistance for a long time. If people questioned it I said, call that number, ask for me, they will tell you the number to call :-)
In fact, you can often keep the activated card (at all kinds of stores) even if they do ask you to fill in some papers. Just ask to take the forms home to fill them in and return later and take the card, too.