I have been offered a place as a PhD student in an English-speaking university, however, I'm a little concerned that the department may be a bit broad and generic (not the project though; it appeals to me greatly and is specific to my interests). It's not a big university, and as a consequence doesn't really have a granular department/group structure.
When it comes to the concept of the work a PhD entails, I understand it well; however, (rather ashamedly) I am questioning my understanding of the relationship between the University name, the school, and the research topic in the formulation of the PhD title, as well as how one might refer to the degree in official situations. It was my understanding that, technically, a PhD should be a standalone demonstration of novel research, and can stand on its own merits without further qualification of which group/department you did your research under, rather than the department forming the basis of the answer to the standard sort of, "What's your PhD in?" questions.
For example, a quite specific biology-related project might be attached to a Biology Department, Computer Science department, or to Engineering etc. depending on the disciplines involved.
Would that individual refer to their study as a "PhD in Biology Department"/"PhD in CS Department" etc. on record, e.g. a faculty page/LinkedIn or would they more likely use "PhD in A quite specific biology discipline" based on the content of their PhD, e.g. viral microbiology or cancer bioinformatics?