This article is from the Credit cards and Consumer Credit FAQ, by firstname.lastname@example.org (Steve Adams) with numerous contributions by others.
Don't rule out ignorance: many merchants don't know the rules.
They may think (wrongly) that getting extra information from you
will protect them somehow. The truth is that if they follow the
procedures of the credit-card company, they will get paid, period.
On the other hand, some merchants are deliberately flouting the
rules and depending on you to acquiesce. Why? Because they can
sell your address or phone number, or add you to their in-house list
of sales prospects.
Don't accept the old wheeze about "in case there's a problem." If
the merchant follows proper procedures at the time of sale, there
won't be. If you leave your card behind they can send it to the the
card issuer, who will return it to you.