What browser engine did Opera use from Opera 7 until Opera 15? Presto.
Presto was the proprietary browser engine used by Opera from the release of Opera 7 in 2003 through Opera 12. Opera 15, released in 2013, marked the transition to Blink, the engine created for the Chromium project. Presto is therefore the canonical answer for the historical span in the question.
Presto powered Opera’s distinctive browser features and was also used in related products, including Opera Mobile and parts of Opera Mini’s browsing architecture. It was designed as a dynamic engine capable of re-rendering all or part of a page in response to document events, and its development included extensive work on standards support and JavaScript performance.
A frequent mistake is answering “WebKit” because Opera announced a move toward WebKit before adopting Blink. Another is choosing an ECMAScript engine such as Carakan, which handled JavaScript within the broader Presto platform. Those technologies are related but are not the browser engine named by the question.