I got what is Theymos's idea but sending a package with just an address and not a name when the package/gift you are receiving is expensive is risky. I wouldn't take a chance of not sending my name
You could use your real name with your real address. That information is probably not very secret anyone for someone who knows your address.
This feature would surely improve privacy on the forum if it is utilized. I have once mentally declined participation in a contest cause it could potentially compromise my identity and location on the forum.
I've never bought anything physical nor joined any physical giveaways because of privacy concerns. Even with this newly changed system I don't think I'll join.