Can one organisation be a Fiduciary and a Processor at the same time?

Yes, and this is common. The role depends on the activity, not the entity. A logistics firm is a Data Processor when it handles delivery addresses on a retailer's instructions, but a Data Fiduciary for its own employee records and its own direct customers. A marketing agency is a Processor for campaign data it runs on a client's behalf, and a Fiduciary for its own contact database. A manufacturer is a Fiduciary for its dealer network data and a Processor when handling warranty data for a brand it supplies. The practical step is to map roles activity by activity rather than assuming one label describes the whole organisation.