Shopify Translate & Adapt (Shopify Inc., Ottawa, Canada) is widely used for store translation. These Shopify apps come from providers based in Europe. What I could document about privacy and hosting is on each detail page.
Paris, France
Translates store, theme and apps automatically and creates separate, SEO-ready language URLs.
Herford, Germany
Translates store texts, images and content from third-party apps, with a language switcher in the theme.
For store translation, the directory lists 2 apps from European providers: Weglot, langify. I checked each provider's location against a source.
No. A European headquarters does not mean all data stays in the EU. Each detail page says what I could document about the DPA, hosting and subprocessors, and what is still open.
Shopify Translate & Adapt is made by Shopify Inc., Ottawa, Canada.