If you buy Bitcoin through P2P, you may use cash, gift cards, or prepaid cards. You may also purchase via over-the-counter with cash. There are also Bitcoin ATMs that only ask for your phone number when buying. You may use a burner number for that. You can also opt for meet-ups in doing transactions. Again, with cold cash, of course.

All of these options would keep your identity private. However, you may appear personally or be recorded in a camera. That doesn't necessarily mean you reveal your identity, though. You can always do something to hide your real face. You can buy a 1$ mustache, for example.