Yes, it is possible to run a Shopify dropshipping business while maintaining anonymity to a certain extent. However, it's important to note that complete anonymity may not be achievable due to legal requirements and the nature of online transactions. Here are some strategies you can employ to protect your identity:

1. Use a business name: Instead of using your personal name, register a business name or use a DBA (Doing Business As) name. This allows you to operate your business under a different identity.

2. Privacy policies and terms of service: Create a privacy policy and terms of service for your website that clearly outline how you handle customer information and protect their privacy. This helps establish trust with your customers and shows your commitment to safeguarding their data.

3. Separate business entity: Consider forming a separate legal entity for your business, such as an LLC (Limited Liability Company), which can help protect your personal information. Consult with a legal professional to understand the best approach for your circumstances.

4. Private domain registration: When setting up your Shopify store, opt for private domain registration. This service replaces your personal contact information in the domain registry database with generic information, keeping your personal details hidden.

5. Use a virtual business address: Instead of using your home address, consider utilizing a virtual business address or a mailbox service. These services provide you with a physical address where you can receive mail and packages, without disclosing your residential location.

6. Professional email address: Create a professional email address using your business name or domain name. Avoid using personal email addresses that may reveal your identity.

7. Limited social media presence: Be cautious about sharing personal information on social media platforms. Create separate business accounts and avoid posting personal information that could link back to your identity.

8. Secure website: Implement security measures such as SSL certificates to encrypt customer data during transactions. This helps protect sensitive information and builds trust with your customers.

9. Third-party fulfillment: Utilize third-party fulfillment services to handle order processing and shipping. This way, your customers will receive packages without your personal information being disclosed.

10. Seek professional advice: Consult with legal and accounting professionals to ensure you comply with local regulations, tax requirements, and any other legal obligations while maintaining your desired level of anonymity.

While these strategies can help protect your identity to a certain extent, it's important to understand that certain information, such as payment processing details, may still require disclosure to relevant parties for legal and financial purposes.