To give more context, I am in the US.
I am a first-year PhD student in CS, and I was admitted without an advisor. Over the first semester, I have talked to many professors to see if our research interests match. Finally I decided to work with Prof. X, and I told him that. He then followed with a plan for a study project related to his research, eg, playing around with his codebase, reading his lecture notes from his previous special-topic course. So far, we have had two weekly meetings, and he suggests we should meet every week and I should email him if I have problems in this study.
In this case, has he considered me as his PhD student, or someone on probation as a candidate? I think of him as my advisor, and I follow what he suggests me to do. But does he think the same way? He has never announced something to indicate our student-advisor relationship. In our meeting, we have only discussed problems on actual contents of the subject, never about something else such as whether to give me an office space, whether to fund me as RA so I do not have to TA.
In my mind the answer I want is whether I am officially a PhD student of Prof.X. But how do I phrase it so it is not too awkward to confirm with him? Or should I just silently believe that it is already the case?
I know right now the project is a non-research study, so should I ask him about office space or RA now or later when I actually start researching?
The department continuously tracks me as "in search of advisor", so I think I am urged to give them an answer. But with my current situation, I do not know what to say to them.