I've sent many items over the last couple years, always with my contact info (phone, email and twitter on the gift card... having them wrapped ensures the info is prominently displayed on a gift tag rather than on the receipt where it is less obvious).
So the answer is, it sort of depends. For anyone doing this, I must warn you that the 'acknowledgement rate' is very very low... like 5-10%. The other 90-95% you know the items were delivered, but you will never hear from the girl. Even via social media. However, on the up side, I have used this process to meet a couple that weren't widely known as available. If you just have to do something nice for your fave, its fine.
As far as Amazon goes, your name won't be included unless you include it. Amazon has many sub vendors, and if the item is shipped from them, typically there will be no information at all as to who it comes from. If it's from Amazon stockpiles, there will be a receipt and you can have the information omitted. Anything you send as a gift you have the option of no receipt sent with the package.
On the extreme down side, I had one very nice item ($700+) that created a situation where the girl needed to provide a confirmed delivery address to an Amazon 3rd party. Because of the 'anonymous' nature of the gift list itself, the 3rd party contacted me instead. I told them they had to contact her. After a couple days the vendor called me back. Apparently the 3rd party thought it was like a 'Make a Wish' type wishlist for kids or something, because she felt like I was being betrayed, as the PS said 'it was for my boyfriend, but we broke up. Can't you just send me the money, because I am going to return it anyway and I know Amazon will give me the credit, so why don't you just save us both the hassle and send the money instead.' Since the order hadn't completely processed on Amazon I cancelled it and got the refund myself. Needless to say she is officially off my Christmas list...
