In what branch of science does technology belong?
Actually, technology per se isn't science. But then, neither is mathematics. Technology is normally about the creation of things. Science is about using a certain methodology for trying to discern the truth about the world and natural processes. If you define natural processes broadly, including human processes, then social science is also science. 

But mathematics is about using logical process to explore the implications of basic assumptions (axioms).

But both science and mathematics are about ideas, whereas technology is about things. 

Technology, as in iPhones and respirators and all, is about the creation of useful things. That is basically why you can patent inventions, but you can only copyright scientific papers. 

Some sorts of technology, say vaccines, depend on science to assure safety and efficacy. And some sorts of technology depend on mathematics, such as computers and bitcoin. 

Computer Science has elements of both science and technology. But one normally gets a PhD in CS for discoveries on the scientific side, even if the science is an exploration of something created on the technological side. CS actually has elements of mathematics as well, such as in algorithm analysis and computability theory.