also, I don't trust them with my DNA data anymore.
Can you use a fake name?
Yes. A friend of mine is currently doing this. They used a fake name and paid with a pre-paid Amex "gift card". They had it mailed to a P.O. Box. I believe they may have used VPN and private browsing while ordering and looking at the website, and obviously created a new email account with hotmail, or whomever.
Now, obviously, you're asking them to sequence your genome, so they're getting personally identifiable information. If you have relatives who have done their 23 and me work-up, then they'll be able to tell that "John Doe" is related to Gunther Kumar, or whatever. In my friend's case, that's not likely to be an issue as they have few living relatives, most of whom are very old and would have no interest in 23 and me. YMMV.