If you're in the EU, you might be able to try some GDPR moves. I'm not so IDK much about it, but there are other things to think about about online buying privacy.

For physical products, they'll need some addr to send to. You can use a PO box if avail in your area. Prob is most sites have an anti-fraud system. The less info it has to pin you to a real world ID, the more likely you'll trigger it.

That happened to me on ebay. Had the best possible rating, literally never got less than perfect feedback in many hundreds of transactions. Despite most of a decade of good faith use, one day they locked my acct. Demanded photo ID to unlock. Which I didn't provide, so it remains locked. I think it was b/c I used a PO box and an alias, and I block lot of browser fingerprinting. That was all fine. Until one day it wasn't.

This is why I hate scammers so gd much. Scammers use the same methods we want for honest privacy, but they use it to scam, which makes co's get super aggressive about ID'ing everbody. Scammers poison the privacy well for everybody.