Data controller — under GDPR Article 4(7), the natural or legal person that "alone or jointly with others, determines the purposes and means of the processing of personal data."
For a Shopify store, the merchant is the controller. Shopify and your installed apps are usually processors (they handle data on your behalf, per your instructions).
Why this matters:
- Customers exercise rights (DSARs, deletion requests) against you, not against Shopify.
- You're contractually responsible for ensuring your processors are compliant — that's why you need DPAs.
- Fines under GDPR are usually levied against the controller; the processor is only directly fined for processor-specific violations (security, sub-processor management).
- If a Shopify app leaks customer data, you're on the hook to your customers; the app is on the hook to you.
See also: GDPR, Data processor.