Yes.
The hotel needs a written authorization from you, with ID, to charge your credit card for another person's stay. Otherwise you could just dispute the charge with your credit card and they would lose money.
On the other hand, I have ordered/paid for quite a few Christmas gifts - being sent to other people - just by giving my card number over the phone...
So are folks saying this fax/ID policy is specific to hotels/SPG?
Cheers,