Amazon doesn't include the name of the buyer in the music file - either in the file's mp3 encoding itself or in the ID3 tag. Tagging the ID3 as you download the file is technically possible, but changing the MP3 encoding on-the-fly as you buy/download the file isn't particularly feasible for several reasons. The most notable is that there would be a 'privacy' outcry from the dark corners of the web, the way they reacted when Apple seemed to do something like that with iTunes.

If you have any suspicion about it, get a friend to buy a random MP3 from Amazon using her account and buy the same file yourself using your account. Compare the two MP3s using whatever tool you want; I did, and they seemed to be absolutely identical.

FWIW, there is a field in the Amazon MP3 ID3 - under 'Comments' - which says Amazon.com Song ID: 20XXXXXXX (I Xed the last seven digits). That's a unique song identifier, not a user identifier.