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).
So. Yeah. I'll pass.